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Tous les autres exempiaires originaux sont film^s en commenpant par la premiere page qui comporte une empreinte d'impression ou d'illustration et en terminant par la derniire page qui comporte une telle empreinte. Un dtis symboles suivants apparaitra sur la demidre image de cheque microfiche, selon le caa: la symbole — ^ signifie "A SUIVRE", la symbols V signifie "FIN". Les cartes, planches, tableaux, etc., peuvent etre filmte dt dee taux de rMuction diffdrents. Lorsqua le document est trop grand pour etre reproduit en un seul cliche, il est filmd d partir da Tangle sup^rieur gauche, de gauche d droite, et de haut en bas, en prenant le nombre d"images n^cdssaire. Les diagrammes suivants iilustrent la mdthode. 1 2 3 4 5 6 TJ Aa PI Serica of National 0cl)ool Book TREATISE ON ARITHMETIC, m THEORY AND PEAC'JICE. ^ I FOR ^%^ UlSit (9(5 ^I^DjlUiVa^. Aathorh-d hy the Council of Public Instruction, for Upper Canada. TORONTO: PUnLISIlI^^D BY ROBERT MclMIAIL, 65, KTiNG Strket East. • 1860. * \ 1 V \ \ \n ti t€ \ ''Vv^^' ij^^ // / I' K E F A U JK > In Iho prosont edition a vast number of eicrcisM I,.,. be,n «ldod, that no ,„,e, however trifling nlTL ,oft w.thout ,0 many ill„,t™tiona as should f rvet ml u uftc,ently fi.^.har t. the pupi,. And when it was feared m-glit not at oneo suggest itself; some question calculated ocrs and " the pnnoiples of notation and numeration - for the teacher may re.t assured, that the facility, I^d e^^ the success, wUh which subsequent pa>-ts of his tostrucZ wirbe .onveycd to the mind of the learner, dependsinl' great degree, upon an adequate acquaintance wHh ti Hence, to proceed without scouring a perfect and pra^S knowledge of this part of the suyect, is to retard'TaS than to accelerate improvement. . ?» '■"P''-.*'™"' "« ^--y oommencemcnt, must be m.>da due d. Of the great utility of teehnica'. iangua..e faecut tately „ndersto«l) it is almost superfluous to say anvZv hero : we cannot, however fni.l,o..,. ^i ■ ' ""y'hmg cillin,, t,,.. • ""we\oi, iMbear, upon this oocas on, re- call ng to ren,cmbra«ee wh.at is so admirably and so effeo. J.ven n the common mchanical arts, something of a t.chn,cal language is 6,md needful f„r th<L. who .arMcaru? '* PREFACE , ing or exercising them. It would be a very groat la convenicnpo, even to a common carpenter, not to have a precise, well understood name for each of the several opera- tiona he performs, such as chiselling, sawing, planing, ^^c, md for the several tools [or instruments] he works with. And If we had not such words as addition, subtraction, multiplication, division, &c., employed in an exactly defined sense, and also fixed rules for conducting tho.e and other arithmetical processes, it would be a tedious and uncertain work to go through even such simple calculations as a child very soon learns to perform with perfect ease. And after all there would be a fresh difficulty in making other per- sons understand clearly the correctness of the calculations made. "You are to observe, however, that technical Ian.rua.^o and rules, if you would make them really useful, must be not only (bshndly urulerstood, but also learned and yemembered as famiharly as the alphabet, and employed cmshmtlu, and With Born^uhns eimtnm; otherwise, technical langua-e will p.-ove an encumbrance instead of an advantage, just as a suit of clothes would b. if, instead of putting thorn on and wearing them, you.were to carry thorn about in your hand " Page 11. What is said of technical lan^riMge is, at least, equally true 0. the signs and characters by which we still further tae'ilitate the conveyance of our ideas on such matters as form the subject of the present work. It is mucli more simple to put down a character whicu expresses a process, than to write the name, or description of the latter, in fuJl. Besides, in glancing over a mathematical investigation, the mind is able, with greater ease, to connect, and understand its dif- ferent portions when they are briefly expressed by familiar Bigns, than when they are indicated by words vdnch havo nothing particularly calculated to catch the eije, and which cannot even be clearly undcrstocKi without considerable attention. But it mu«c be bortie in mind, that, while such «■ trsatise as the present, will seem easy and intelligible ▼a PREFACE enough if M,e „ign,, ^Uch it contains in almost evorr na™ mo or loss obscure to those who have not been habitfS t he „so them. They are, however, so few and so L^H that there is no eicnse for their not hoin^ perfcctlv onir stood-partienlarly by the teacher of arithme:;o ' '''■ hhould peculiar e.rcumstanccs render a different arrange- ■ IT ' u ° ■"■''"'"' "' ■«"^'' ""i^^aWe, the j„dicio,« master will never be at lo8« how to act^there «« Z ntelhgcnco of tho pupil, will render it necessary to conflno u .nstrucfon to the more important branches. 4, i^^r Bhould, If possible, mako it an inviolable rule to receive rerorVetr """"'"ri' "^ "^ e^pWion r reason. The references which have been subjoined to tha different questions and which indicate the paJgiphs wh re neswrh "'' ''"^*'" be obtained, and alsfthL^r enees which are scattered throagh the work, will, be fould of considerable assistance ; for, as the most i^teUi-nt pnTu wi 1 oecasionally forget something he has learned he^a^ e.ple" ■ ""' '™ '»™"*^' *'"'™ '"e has seen it I)«™<,& have been treated of at the same time as integers be^^ause since both of them follow ,re,set, the same ifw"' when the rales relating to integers are fullv understood here IS „o&V,. „» to be learn:! on the sub/eelrpt^ ct larly If what has been said with reference to numeratirand notation is carefully borne in mind. Should it, however "n unti the learner shall have made some further advance. _ ilie most useful portions of menlai arithmetic have been ntrodueed into "Practice- and the other rules with whch they seemed more immediately connected. the mind of the toner and when he is found to have been ^" PRKFACE. guilty of any inaccuracy, ^o Bhould be made to correct him »elf by repeating each part of the appropriate rule, and oxeniphfyin- it, until ho perceives hia error. It should be continually kept in view that, in a work on such a subject as arithmetic, any portion must seem difficult and obscura without a knowledge of what precedes it. The table of logarithms and article on the subject, also the table of squares and cubes, squarr. roots and cube roota of numbers, which have been introduced at the end of the work, will, It IS expected, prove very acceptable to the more •dvanced arithmetician. n CONTENTS f PART I. Multiplication Table, . . . , Tables of money, weights, and mcasuree, Definitions, • . . . Section I.— Notation and Numeration, Arabic system of numbers, lionian notation, . ' • • • Skction II. -^Simple Addition, To prove Addition, Addition of quantities containing decimals, Simple Subtraxjtion, To prove Subtraction, iSubtraction of quantities containing decimals, Simple Multiplication, . To multiply when neitlier multiplicand nor mul tiplier exceeds 12, . When the multiplicand exceeds 12, To prove Multiplication, To multiply quantities, when there are cyphers or decimals, When both multiplicand and multiplier ex eee«^. 12, To prove Multiplication, To prove MultipHcation, by casting out the nines, To multiply, 60 as to have a certain number of decimal places in the product, To multiply by a composite number, ~ by a number not composite. To multiply by a number consisting of nines. Simple Division, . . a 2 Par* 1 8 12 14 18 20 83 37 45 It) 41) 53 57 57 CO Gl 6r> G7 70 72 73 74 78 I'll CONTENTS when tho divisor does not exceed 12, Mividend 12 times the divisor, ♦1, V .J ".'" *^'^'^°'" ^*^* ""* "oeed 12, bufc 4:ot:s:r"*^^^ To divide when the dividend, divisor, or both' contain cyphers or decimals, - When the divisor exceeds 12, To prove Division, . * ' To divide by a composite number, ^^^^^ ^L^ "'""^'''^ ^"- ""^« J^« *han one expressed by unity and one or more cyphers, lofind the greatest common measure of nimberi 10 tnd their least common multiple SECTION III.-Reduction Descending, ' ' Reduction Ascending, . . * * * To prove Reduction, . ^,' The Compound Rules, . .' * ' ' Compound Addition, . ,' ' ' Compound Subtraction, Compound Multiplication, when the multiplier does not exceed 12, . . • When the multiplier exceeds 12 and ig composite, . . «"iu is — - When the multiplier is the sum of compo^ Bite numbers, . . ^ -— When the multiplier is not composite, .' Compound Division, when the divisor la abstract, and does not exceed 12, . ■ When the divisor exceeds 12 and' is com. pgsite. When the divisor exceeds 12 and is not com posite, -— When the divisor and di vid'end are both apl plicate, but not of the same denomination • or more than ono 'lenomination is found in either or both, 80 85 86 89 94 05 98 90 lOZ 104 lOV 109 110 113 114 123 126 128 123 130 132 • 134 134 139 ^ •i 80 85 86 89 94 9S 96 96 lOZ 104 lOV 109 110 113 114 123 ^ // { /. COMTKNTi /v* / %(:tion IV.— Vulgar Fractions, \ Toreducoun improper fraction to a mixed n To reduce un integer to a fraction, To reduce fractions to lower terms, To find the value of a fraction in terms of a lower denoiaination, . To express one quantity as the fraction of another, To add fractions having a common denominator, To add fractions when their denominators are different and prime to each other. To add fractions having different denominators, not all prime to each other. To reduce a mixed number to an improper frac- tion. To add mixed numbers, . ' • • To subtract fractions which have a commoh denominator, . ■ • • • To subtract fractions which have not a commoa denominator, . . . . , To subtract mixed numbers, or fractions from mixed numbers, To multiply a fraction and whole number together To multiply one fraction by another, To multiply a fraction, or mixed number by mixed number, To divide a fraction by a whole number, To divide a fraction by a fraction, To divide a whole number by a fraction, To divide a mixed number by a whole number or fraction. To divide an 'nteger by a mixed number, To divide a fraction or mixed number by a mixed number, ' • • • ►Vheu the divisor, dividend, or both, are com. pound, or complex fractions, . Oeoimal Fractions, .... To reduce a vulgar fraction to a decimal or to a decimal fraction. To reduce a decimal to a lower denomination, /140 ^ 142 142 143 143 144 145 146 147 148 149 150 150 152 153 154 156 156 158 158 159 159 160 162 163 163 CONTENTS. To find at <me the decimal equivaJent to any number of shillings, pence, &o., ^ Circulating Decimals, . * * * To change a circulating decimal into its equi* vaJent vulgar fraction, ^ When a vulgar fraction will give a finite decimal,' Thenumber of decimal places in a finite decimal The number of digits iu the period of a circulate — When a circulating decimal will contain a hnite part, . . Contractions in multiplication a«d division, de rived from the properties of fractions, Section^ \.— Proportion, . Natiu^ of ratios, . , '' Nature of Proportion, To fil *^' ^jthmetical mean of* two quantities,' To find a fourth proporti. al, when the first term isumty, . . ^ "* — When .ho second or third term is imity, To find the geometrical mean of two quantities,- Fr operties of a geometrical proportion, . Rule of Simple Proportion, — When the first and second terms are not of .he sa:ne or contain different denominations, — When the tbu-d term contains more than one denomination, ... If fractions or mixed numbJrs are found in any of the terms, Piule of Compound Proportion, To abbreviate the process, ' . PART n. &CTI0N VI.~Practice, To find aliquot parts, . . * * To 2nd the price of one denomination, ttiat of a higher being given, , rtfft 164 im 169 167 170 170 171 172 173 176 17T 179 181 184 184 184 185 185 190 192 195 202 204 203 209 21] 1 J raff* iny • 1Q4 mt • 169 • 169 • a- • 167 il, 170 il, 170 e, 171 a • 172 9- • 173 • 176 • 17r k 179 >, 181 1 • 184 • 184 184 1 185 185' 190 192 195 202 204 209 21] CONTENTS. To find the price of more than one lower denomi- nation, • . . , To find the price of one higher denomination, .' To find the price of more than one higher deno- mination, Given the price of one denomination, to find that of any number of another, . When the price of any denomination is the aliquot part of a shilling, to find the price of any num- ber of that denomination, When the price of any denomination is the aU- quot part of a pound, to find the price of any number of that denomination, . When the complement of the price, but not the price itself, is the aliquot part or parts of a pound or shilling. When neither the price nor its complement is the aliquot part or parts of a pound or shilling. When the price of «ach article is an even number of shiliin./A to find the price of a number of ai'ticieK * • • • When the price is an odd numbor and loss than 20, . . To find the price of a quantity reproscited by a mixed number, Given the price per cwt., to find that of cwt., qrs., &c., Given the price per pound, to find that of cwt.,* ^qrs., &c., Given the price per pound, to find that of a ton. Given the price per ounce, to find that of ounces, pennyweights, &c., . Given the price per yard, to find that of yards,' qrs., &«., To find the price of acres, roods, &o., . Given the price per quart, to find thet of a hoc*- tead. ". Given the price per quart, to find that of a tun, ziu Page 212 213 213 214 215 215 216 217 218 219 220 221 222 223 223 225 226 226 227 xiy CONTENTS Given the pri^e of one article in pence, to find that of any number, . Given wages per day, to find their amount per year, . . ^ ^ . Bills of parcels, . . ; • . Tare and Tret, . ' * * • • • . Section Vll.-Simple Interest. To find the simple interest on any sum, for a year, — When the rate per cent, consists of more than one denomination, To find the interest on any sum for years* For years, months, &o., To find the interest on any sum, for a^y time," at .5, b, &c., per cent., . When the rate, or number of years, or both,' are expressed by a mixed number. To find the interest for days, at five per cent., .* To find the interest for days, at any other ratq, . 10 find the interest for months, afr 6 per cent , To find the interest of money left after one or more payments. Given the amount, rate, and time~to find the principal, Given the time, rate, and principal-to find the amount, Given the amount, principal, and rate— to find the time. Given the amount, principal, and time,' to find the rate. Compound Interest-given the principd, rate,' f '.d time— to find the amount and interest. To find the present worth of any sum. Given the principal, rate, and amoJnlrlto find the time. Discount, • • • To find discount, • , , To compute Commission. lnsiirnTift« 'p««i,^-„«» Pajf« 227 223 229 233 237 238 239 239 240 i 242 1 243 i 244 M 244 i 246 1 248 • i^?M 249 1 249 1 250 1 251 1 256 1 257 1 260 ^ 261 1 nan li Pagf« 227 223 229 233 237 238 239 239 240 242 243 244 244 246 248 249 249 250 251 256 257 260 261 nan CCNTENT8. To find what insurance must be paid that, if the goods lire lost, both their value and the insur ance paid may be recovered, .. Purchase of Stock, Equation of Payments, . Sjcction VIII.— -Exchange, . Tables of foreign money. To reduce bank to current money, To reduce current to bank money. To reduce foreign to British money, To reduce British to foreign money, To reduce florins, &c., to pounds, &c., Flemish, To reduce pounds, &o., Flemish, to florins, &c., Simple Arbitration of Exchanges, Compound Arbitration of Exchanges, To estimate the gain or loss per cent.. Profit and Loss, . • • • To find the gain or loss per cent.. Given the cost price and gain— to find the selling price, . . . . ^ Given the gain or loss per cent., and the selling price*-to find the cost price, . Simple Fellowship, Compound Fellowship, . . , ' Barter, Alligation Medial, Alligation Alternate, When a given amount of the mixture is re quire,d, When the amount of one ingredient is given. Sjcction IX.— Involution, . To raise a number to any power. To raise a fraction to any power. To raise a mixed number to any power, Evolution, To find the square root, . When the square contains decimals., To find the square root of a fraction, XV Page 263 265 266 268 269 272 273 274 277 281 282 283 285 286 287 288 289 290 291 293 296 298 299 302 304 306 307 307 308 308 308 '^U M2 XVI CONTENTS. I To and the square root of a miiod number, 10 IincI the cube root, 'T7 Y^®" *^^ ^^^^^ contains decimals, ' lo find the cube root of a fraction. To find the cube root of a mixed number,* 1 extract any root whatever, To find the squares and cubes, the squm-o and cube roots of numbers, by the table, . Logarithms, . \tle, ^^' ^'^^''^^'^ "^ ^ given numbed by th To find the logarithm of a fraction. To find the logarithm of a mixed numbed, log^iim, "^"'^^ ^^^;'^^'^^"^^"« *« ^ Siven — If the given logarithm is not in the' table, * lo multiply numbers by means of their loga-* To divide numbers by melns of their logarithms* l^^t^^^^^^^^^P--^^-^^ To evolve a quantity by means ofits logarithm. ' SECTION X.— On Progression, To find the sum of a series of terms in arUhmeti-' cal progression, • In an arithmetical series given the extremes and number of terms-to find the common diS ence, , ^ \^.t r """"^'^ '^ arithmet'ical m;ans,be: tween two given numbers, ^'s fries,'"^ ^''''^'"^'' '"'"^ '^ '-^"y '^^^'itl^neticai ^"cZ&'7-f'"^ ''"'^' *S™ '^' ^^*^-«*"^- and common difference-tofindthenumberof terms, ~r. '\*^' '"™ "^ "^« ««^i««' "^e number of Tn ni' ''"" «^treme-to find the other, . To find the sum of a series of terms in geometri- oal progression, . b^ometri- rage 313 313 315 316 31G ' 317 318 319 321 323 323 324 324 325 326 327 327 329 329 330 ooi 332 332 333 CONTENTS. Ih a geoulGtrical series, given the extremes and number of term&— to find the common ratio, To find any number of geometrical means between two quantities, To find any particular term of a geometrical series, In * geometrical series, given the extremes and common ratio— to find the number of terms, . In a geometrical series, given the common ratio, the number of terms, and one extreme— to find the other • . . . Annuities, . . * ■ • To find the amount of a certain number of pay- ments in arrears, and the interest due on them, To find the present value of an annuity, When it is in perpetuity, . To find the value of an annuity in reversion, Position, . 5"ingle Position, . Double Position, . Miscellaneous exercises, Table of Logarithms, Table of squares and cubes, and of square and cube roots, «... Table of the amounts of £1, at compound interest] Table of the amounts of an annuity of £1, Table of the present values of an annuity of ^1, Irish converted into British acres, ' Value of foreign money in British, . [ Tago 335 336 336 337 338 340 340 342 343 344 345 346 347 355 361 377 385 385 386 386 386 T5/ EATLSi: ON ARITHMETIC: IM THEORY AND PRACTICE. ARITHMETIC. PART I. TABLES. MULTIPLICATION TABLE. |i Twice 1 are 2 2—4 3 *- 6 4-8 5 — 10 6 — 12 7 - 14 8 — 16 9 — 18 10 — 20 11 — 22 12 — 24 8 times 1 are 3 2—6 8—9 12 15 18 21 24 9 — 27 10 — 80 n — 83 12 — 36 4 times 1 are 4 2—8 3 -r- 12 4 — 16 5 — 20 6—24 7—28 8—32 9 10 11 12 86 40 44 48 5 times 1 are 5 2 — 10 3 — 15 4 — 20 6 — 25 6 — 30 7 — 35 8 — 40 9 — 45 10 — 60 11 — 55 12 — 60 6 times 1 are 6 2—12 3 — 18 4 — 24 6 — 30 6 — 36 7—42 8 — 48 9 — 54 10 — 60 11 — 66 12 7'^ 7 times 1 are 7 2 — 14 8—21 4 — 28 5—35 6—42 7—49 8 — 66 9 — 63 10 — 70 11 — 77 12 — 84 8 times 1 are 8 2 — 16 3 — 24 4 — 32 5 — 40 6 — 48 7 — 56 8 — 64 9 — 72 10 — 80 11 — 88 12 — 9S 9 times 1 are 9 2—18 3—27 4—36 6—45 6—64 7 — 63 8—72 9—81 10 — 90 11 — 99 12 — 108 10 times 1 are 10 2—20 3—30 4—40 6—60 6 - 7 - 8 - 9—90 10 — 100 11 — 110 12 — 120 60 70 80 11 times 1 are 11 2—22 3—33 4—44 5 — 55 6—66 7—77 8—88 9—99 10 — 110 11 — 121 12 — 132 12 times 1 are 12 2—24 3—36 4 — 48 I 6 — 60 I 6 — 72 I 7 — 84 , 8 — 96 i 9 — 108 ! 10 — 120 \ 11 — 132 12 — li4 It appears from tliis table, that the multiplication of tho same two uumbers, m whatever order taken, produces tlio SIGNS USED IN THIS TREATISE. ' 7 times i 1 are 7 2 — 14 3 — 21 4 — 28 5 — 35 6 — 42 7 — 49 8 — 66 9 — 63 10 — 70 n — 77 12 — 84 12 times 1 are 12 2 - 24 3 - 86 4 — 48 6 — 60 e — 72 , 7 - 84 . 8 — 96 ; 9 ~ 108 ! — 120 I 1 — 132 2 - Ui ion of tho iucca tlw ^^ + the sign of addition; as 5+7, or 5 to bo addc4 tra7ted''froT4"^ "*'''°"™ ' "^ ^-^' "' ^ '» ^' »»''. muHiptVbfg"' -"'"J"''''"'-'" i »» 8X9, or 8 to be ^^•^ the sign of division ; as 18+6, or 18 to be divided ti,„ ft? vinculum, which is used to show that all the quanfafe s united by it are to be oonsiderld as but rbe^^n^r^^^^I^l-ral^e^-f^p^HeltJ thl|ts'lTssfh™*i.""'" "■"* ' '^ S>-«^'er than J, and thJ rati^of frZ/'i"- '" '■f"".™' *"' 5:6, means me ratio ot j to 6, and is read 5 is to 6 : : indicates the eqiiality'of ratios ; thus, 5 : 6 • • 7 • 8 means that there is the slm. relation bet'ween 5 'and 6 as between 7 and 8 ; and is read 5 is to 6 as 7 is ?o 8 y the radical sign. By itself, it is the sign of the 2:\t rootff 4', or 4^ it "^^ ^' '' '' '' ^^' '^ 641-31, &c. may be read thus : taE 3 f?om 8, add 7 to thJ difference, multiply the sum by 4, divide the nroduct hv fi take the square root of the quotient Td to °t ^dd 31 tLn multiply the sum by the cube root of 9, divide tho product ^ — r' — '•' ••'- product will bo equal to 041-31. &c 1 hesc «sns arc /tUly cq^laincd in tla-ir proper places. I MULTIl'LICATIOiN TAIJI.E. eaino result; tlius I times 0, and G times 5 jiro 30:— tho reason will bo oxplainod when we treat of multiplication. Ihero are, therefore, several repetitions, which, although many persons conceive tlioin unnecessary, are not, perhaps, quite unprofitable. Tho following is free from such an objection : — f Twice 2 » 3 4 6 6 7 8 9 >> >> •> •> II II are 4 — 6 — 8 ~ 10 ~ 12 — 14 — 18 — 18 8 times 8 4 6 6 7 8 9 II II II II II II 9 12 15 18 21 24 27 6 times 7 are 35 8 — 4a 9—45 II 6 times 6 7 8 9 II II II 86 42 48 64 4 times 4 6 6 7 8 II •I II II 16 20 24 28 82 86 5 times 6 — 26 II 6—30 7 times 7 II 8 1, 9 — 49 — 66 — 63 8 times 8 ., 9 — 64 — 72 9 times 9 81 10 times 8 are 80 II 9 — 90 I, 10 —100 „ 11 —110 10 times 2 are 20 — 80 — 40 — 60 6 — 60 7—70 8 4 6 11 times 2 8 4 6 6 7 8 9 II II II II II II 11 22 83 44 55 66 77 88 99 12 times 2 — 24 „ 8 — 86 4-48 5 60 6 - 72 7—84 8—96 9 —108 10 —120 11 —182 12 —144 I II II II II II "Ten," or "eleven times," in the above, scarcely requiion to be committed to memory; since we perceive, that to multiply a number by 10, we have merely to add a cypher to tho right hand side of it :— -thus, 10 times 8 are 80; and to multiply it by 11 wo have only to set it down twice :-~thus. 11 times 2 rtro 23. TADLE OF MONEY. a 2 — 22 8 — 83 i — 44 5 — 55 3 — 66 J — 77 B — 88 ) — 99 2 — 24 J — 86 t - 48 ) 00 5 - 72 ' - 84 5 — 96 > —108 > —120 — 182 1 —144 7 requirfl.i 3, that to cypher to 0', and to se : — thus, Tho following tables aro required* for reduction, tho compound rules, &c., and may bo committed to memory us convcmonce suggests. ^ TABLE OF MONEY. A farthing ia the smallest coin generally used in this country, it is represented bj . . . /""*"'^' raako 1 halfpenny, i 1 penny, d, 1 shilling, » Kaitlilngg o 4 or 48 900 1,008 halt'ponco 24 or 480 504 pence 12 I shillings 240 or 20 252 or I 21 1 pound, 1 guinea. The symbols of pounds, shillings, and ponce, are placed over the numbers which express them. Thus 3 14 o" means, tliree pounds fourteen shillings, and sixpence'.' So'ilic! times only the symbol for pounds is used, an'd is placed hofore the whole quantity ; thus, £3 „ 14 „ G. 3 9^ moans t ireo 8.nlhno-8 and mnepence halfpenny. 2s. 6?./. means two faulhngs and sixpence three farthinf^n, &c When learning the aI)ove and foflowing tables, the pupil HhMuld be ri^uired, at hr.st, to commit to memory only those pn-t urns which are over the thick angular linos; thusf in tho one just given:— 2 farthings make one halfpenny; 2 half- pence one penny; 12 pence one shilling; 20 shillings one pound; and 21 shillmgs one guinea. a ° Ih h I'oaUy mean the quarter, half, and three quarters of a penny, d. is used as a symbol, because it is the first S. 2 5 13 6 make one half Crown. one Crown. 4 one Mark. I 4 WEIflHTa. AVOIRDUPOISE WEiailT. Its name ia derived from French-— and ultimately from Latin words signify innj " to have weight." It u used in weighing heavy articles Drams 10 ... . 2oO ar 7,168 28,672 448 or 1,792 673,4401 85,840 ouncei 16 pound* 28 112 or 2,240 quarteri 4 SymboU make 1 ounce, oz. . 1 pound, lb. . 1 quarter, q. lhuudrc(l,cwt. hundred! lA IK ■^- 80 or I 20 .1 ton. t. J4 lbs., and m Bomo cases 16 lbs., make 1 stone. 20 stones . . . 1 barrel. TROY WEIGHT. It is so called from Troyes, a city in France, where It was first employed ; it is used in philosophy, in weighing gold, &c. Graini » Symboh. OA •••... grs. ^* • • niftke 1 pennyweight, 'wt. pennyweights 480 or 20 . . i ounce, o%. I ounces 6,760^ 240 or I 12 . 1 pound, . lb. A gram was originally the weight of a grain of corn, taken from the middle of the ear; a pennyweight, that of the Sliver penny formerly in use. APOTHECARIES WEIGHT. In mixing medicines, apothecaries use Troy weight, but subdivide it as follows : — Grains „ . , n(\ Symbolt ^^ ' ' - ^ » make 1 sc< -ipk, ^1 CO or scruples 3 • 480 24 or 288 drams 8 5,760 90 or • • • 1 Uium, 6 1 ounce, 5 ounces 12 . 1 pound, lb. its Carat," which is equal to four grains, is used in wo.iocMg di«monda. The term carat is alf.o applied in o^tiin:;nM>; Jie fineness of gold ; the latter, when ju.vf^nti,/ .♦ MBASU1RES. puro, *i8 said to bo " 24 oarata line." if there are 23 parta ^oid, and one part gomo other material, the mixture is said to he "23 carats fine ; " if 22 parts out of the 24 are gold, it is " 22 carats fine," &c. ; — the whole mass is, in all caflos, supposed to be divided into 24 parts, of which the number consisting of j»old is specified. Our gold coin is 22 carats fine; puro gold being very soft would too soon wear out. The degree of fineness of gold articles is marked upon thera at I ho Goldsmith's Hall; thus wo generally perceive " 18" on the cases of gold watches; this indicates that they are " ly carats fine" — the lowest degree of purity which is stampcfd. JM. _ ^ 80 An avoirdupoise ounce . 437^ A Troy pound . . 5,760 An avoirdupoise pound . 7,000 A Troy poimd is equal to 372- 9G5 French grammes. 175 Troy pounds are equal to 144 avoirdupoise ; 175 Troy are equal to 192 avoirdupoise ounces. CLOTH MEASURE. 24 • • nails 9 or 4 36 16 or 27 12 or 45 20 or 54 24 or make 1 nail. quarters 4 8 5 6 1 quarter. 1 yard. 1 Flemish cU 1 English ell. 1 French clI Linas 12 . ». • 144 or inches. 12 • • ^432 36 or 198 252 7,920 10,080 8o!640 feet 3 • 2,376 3,024 lOior 21 or 660 840 5,280 6.720 yards 7 95,040 120,960 220 or 280 or 1,760 2,210 7r.().?.20 1167^680 LONG MEASURE. (It is used to measure Length.) «. make 1 inch. perches 40 40 32 on ^- 20 or . 1 foot. 1 yard. 1 English perch 1 Irish perch. 1 English fur long 1 Irish furlong. a 1 Vn-liHl. '"!!« .i^ts^sLjLttrttx ticf 2'9h 8 1 Irish railc. »? 6 MEASURED 3 iiiclies 3 palms 18 inches 6 f<^.et 6 feet 120 fathoms 3 palm. 1 span. 1 cubit 1 pace. 1 fathom. 1 cable's length. rnHpr rii'?/- ^ . . ^ ^"^^ a^e equal to 14 Endisli miiec. Ihe Pans foot is eaual tn 19-7Q0 v^ vl ."&"^^* the Roman foot to 11-604 liTih^ ^ u ^""^^'"^ '"^^««J luuL 10 ii OU4, and the French metre to 39383. , MEASURE OP SURFACES foot, a surface one foot long and one foot m^L', L.^ Square inches 144 1,996 3&,304 63,604 ^,668,160 9,640,160 6,272,640 10,160,640 ncake 1 «q. foot. 1 square yard. 10,890 17,640 {43,560 70,560 1 «q. En, 1 sq.Iris porcli.' perah. 4,014,489,600 27,878,400 6,602,809.600i25. 158,400 1 sq. Eng. rood. ' • «q. Irish rood. 1 statute acre. 1 plantation acre. 3,097,600 102,400 5,017,60oll02,400 1 sq. En, 1 sq. Iris mile, mile. rp, „ ' ■ • "'" 1 sq. irisn mile. crXfaff 'aoTmo^ 1'or" ^"-ejarda, and the Irish. 12lVre Irish aero, ' l"" «1™'« Kngibi are equal to anl'The^lg' IKoS'-'^o'^T^'fJ'"'" ^"-, ^-^^ en„o] f,. 101 r-.:-!.'. ' ,' "^"^^ J^^nglish square milfis a^a MKASURES. MEASURE OP SOLIDS. ^ The teacher will explain that a cube is a solid having six equal square surfaces; and will illustrate this by models or examples — the more familiar the better. A cubic inch is a solid, each of whoso a^x sides or faces is a sqitare inch ; a cubic foot a solid cooh of whose osi sides is a square fool ^ &c. Cubic inches I'^^^S . . , . . niftka 1 cub"? fot^vV I cubic feet 27 . . . icubu5r»J4\ WINE MEASURE. Gills or naggina 8 or 32 320 576 1,344 2,016 2,688 pints ') ^ m % • • quarts 8 or 4 • • 3 • gallon 80 40 or 10 « • 144 72 18 • « 836 168 42 . • 504 252 63 • • 672 336 84 • hogshe • Elds 1,008 2,016 504 1,008 126 or 252 2 • 4 or xnai^^ i <«i».*. 1 gaJoiX 1 anker. 1 runlet. 1 tierce. 1 hogsheaa 1 punclieon pipes 2 4,032 8,064 in some places a gill is equal to half a pint. Foreign wines, &c., are often sold by measures differing from the above. 1 pipe or butt 1 tun. ALE MEASURE. Gallons 8 16 or firkins 2 82 4 or 6 8 12 kilderkins 2 48 64 96 8 or 4 or 6 or barrels u make 1 firkin. 1 kilderkin. 1 barrel. 1 hogshead. 1 puncheon. 1 butt. MEASURES. BEER MEASURE. make 1 firkin. 1 kilderkin 1 barrel. 1 hogshead. 1 puncheon. 1 butt. Pints 4 or 8 16 64 152 256 576 DRY MEASURE. (It is used for wheat, and other dry goods.) quarts 2 512 2,048 2,500 4 or 8 32 96 12V. 288 2uG 1,021 1,280 pottles 2 5,120 2,560 4 or 16 48 64 144 128 gallons 2 512 640 1,280 8 or 24 32 72 64 256 320 640 pecks 4 12or 16or 36 or 32 128 160 bushels 3 4 9 320 80 8 or 32 40 coombs 2 make 1 pottle. 1 gallon. 1 peck. 1 bushel. 1 sack. 1 coomb. 1 vat. 8 or lOor 20 quarters 4 5 1 quarter. 1 chaldron 1 wey. weys 10 or I 2 Hast. riurds 60 TIME. MEASURE OE TIME. or 3600, 216,000 5,184,000 t 0,288,000 . 15,152,000 1,892,160,000 1,897,344,000 1,892,160,000 seconds 60 3600 or 86,400 604,800 2,419,200 31,536,000 31,622,400 31,636,000 minutes 60 1,440 or 10,080 40,320 625,000 627,040 525,600 hours 24 Ifi'^ 01 672 01 8,700 or 8,784 01 8,760 Jays 7 28 305 366 365 or SytiaoU make 1 second " 1 minute « 1 hour h. 1 day d 1 week w. 1 lunar month. 1 common year 1 leap year. calendar mon.") 12 I , lunar months f * y^^^' 13 J The following Avill exemplify the use of the above symbols : — - The solar year consists of 365 d. 5 h. 48' 45" 30'": read » three hundred and sixty-five days, five hours, forty-eight minuteB. forty-five seconds, and thirty third;?. The number of days in each of the twelve calendar months will be easily remembered by means of the well known lines, "Thirty days hath September, April, June, and November, February twenty-eight alono And all the rest t)iirty-»ne." The follomng table vrill enable us to find how many days there are from any day in one month to any day in another. From any Day in j T. M >• Cl >• o H i Jan. Feb. Mar April May June July Aug. Sept. Oct. Nov 1 Dec Jan. 36.5 334 306 276 245 214 184 163 122 92 61 31 Feb. 31 36.') 337 306 270 304 246 215 184 153 123 92 62 90 Mar. 59 2S 366 334 273 243 212 131 151 120 April 90 69 31 365 .335 304 274 243 212 182 151 121 May 120 89 61 30 365 334 304 273 242 212 181 161 June July 151 181 120 92 61 31 365 336 304 273 243 212 182 150 122 91 61 92 30 6) 366 334 303 273 242 212 243 Aug. 212 181 153 122 31 365 334 304 273 Sei)t. 243 212 184 153 123 92 62 31 365 335 304 274 Oct. 273 242 214 183 153 122 92 61 30 366 334 304 335 Nov. 304 273 245 214 1.4 153 123 92 61 31 365 Dec. 33^1 303 275 211 214 183 163 122 91 61 30 366 I 10 TIME. iH plucod, and at the samJSf.' i *^^\^«^d of which March ihi left hand sSe of wWch ? On nT^ ^^' horizontal row at intersection the numW o]] • t '"' "^^ P"""^^^^ ^'^ ^^^^r tervene between ti el^fceenth 'oT M T clays therefore, in- October. But thrfourth nf o!. k '"' ' ^?^ *^^ fifteenth of than the fifteenth Zv?, 5/^ *"^T '" ^^^^«" d^y« earl-'er obtjdn 20i'S::r!^^^:^ ^^'^^^^^ " ^-- ^U, and ^f^^^^^^ the bofore in tlie table, wo find tint Tin^ • ? Looking as th., third of J„„„,,y and tt th rd "ffc"t7rt^'"'-^''" tucntli IS sixteen days later thnn *!,„.;■ J' "'f' ""' "'"e- «nd obtaii, 136, the Jiit, required ■*' ™ ''^<' ^« '» ^^^ «Ud one to the 130, aXl'sV tuldtZlt^'r "^ "'"'"^ 'w this bclZTnZl^lTtTl ^"- "■«J"«tand tbe Julian Calenda,;S !"«',,» T '" '."'""^ """ fourth year to the mojrirSarf tftM J P • ''^ Gregory, t.ro..d, this: „..dai™r«,s^;:;::;„„^r TIME 11 to the Julian style, would have been the 5th of October 1582, should be considered as the 15th ; and to preveni' the recurrence of such a mistake, he desired that, in place of the last year of every century being, as hitherto, a leap year, only the last year of every fourth century should be deemed such. The " New Style," as it is called, was not introduced into England until 1752, when the error had become eleven days. The Gregorian Calendar itself is slightly inaccurate. To find if any given year be a leap year. Tf net the last year of a century : Rule. — Divide the number which represents the given year by 4, and if there be no remainder, it is a leap year. If there be a remainder, it expresses how long the given year is after the preceding leap year. Example 1.— 1840 waa a leap year, because 1840 divided by 4 leaves no remainder. Example 2. — 1722 was the second year after a leap year, because 1722 divided by 4 leaves 2 as remainder. If the given year be the last of a century :. ivULE.— Divide the number expressing the centuries by 4, and if there be no remainder, the given one is a leap year ; if there be a remainder, it indicates the number of centuries between the given and preceding last year of a century which was a leap year. Example 1. — 1600 was a leap year, because 10, being divided by 4, leaves nothing. Example 2.— 1800 was two centuries after that last year of a century which was a leap year, because, divided by 4, it leaves 2. • DIVISrW OF THE CIRCLE. Thirds 60 8600 or 216,000 77,760,000 seconds 60 8,600 or 1,296,000 minutca 60 make 1 second " 1 minute ' 1 degree ° I degrees 360 1 circumference. . Ev«»vy circle is supposed to be divided into the same P'^.'^^r of degrees, miautos, &c. ; the greater or less, there 12 DEFlNrnONf?. fore, the circle, the greater or less each of these will be. The following will exemplify the applications of the symbolej : — 00° 6' 4" 6'" ; which means sixty degrees, five minutes, four sccondj, and six thirds. DEFINITIONS 1. Arithmetic may be considered either as a science or as an art. As a science, it teaches the properties of numbers ; as an art, it enables us to apply this know- ledge to practical purposes ; the former may be called theore(tical, the latter practical arithmetic. ' 2. J. Unit^ or as it is also called, Unity ^ is one of tho indivdduals nnder consideration, and may include many units of another kind or denomination ; thus a unit of the order called " tens" consists of ten simple units. Or it may consist of one or more parts of a unit of a higher denomination ; thus five units of the order of " tens" are five parts of one of the denomination called " hundreds ;" three units of the denomination called " tenths" are three parts of a unit, which wo shall presently term *the " unit of comparison." 3. Numler is constituted of two or more units ; strictly speaking, therefore, unity itself cannot be con- sidered as a number. 4. Abstract Nwmhers are those the properties of which are contemplated without reference to their appli- cation to any particular purpose — as five, seven, &c. ; abstraction l)eii^ a process of the mind, by which it sepa- rate^/ considers those qualities which cannot in reality exist by themselves ; thus, for example, when we attend only to the length of anything, we are said to abstract from its breadth, thickness, colour, &c., although these are necessarily found associated with it. There is nothing inaccurate in this abstraction, since, although length cannot exist without breadth, thickness, &c., it has pro- perties independent of them. In the samo way, five, seven, &c., can be considered only by an abstraction of the wind, as not applied to indicate soiae particuiar things. 5. Ajjplicate Numbers are exactly the reverse of DICFINITIONS. 13 (rill be. The ! symbolej : — aiuutes, four IS a science roperties of this know- y be called i one of tlio elude many i a unit of e units. Or of a higher " tens" are hundreds ;" ;enth8" are ly term 'the lore units ; lot be con- aperties of their appli- even, &c. ; tich it sepa- t in reality Q we attend to abstract ough these e is nothing ugh length it has pro- , five, seven, tion of the xliii' things, reverse of abstract, being applied to indicate particular objects — as five men, six houses. 6, The Unit of Comparison. In every number there is some unit or individual which is used as a standard : this we shall henceforward call -the " unit of comparison." It is by no means necessary that it should always be the same ; for at one time we may speak of four objects of one species, at another of four objects of another species, at a third, of four dozen, or four scores of objects ; in all these cases four is the number contemplated, though in each of them the idea conveyed to the mind is different — this difference arising from the different standard of comparison, or unity assumed. In the first case, the " unit of comparison" was a single object ; in the second, it was also a single object, but not of the same kind ; in the third, it became a dozen ; and in the fourth, a score of objects. Increas- ing the '' unit of comparison" evidently increases the (Quantity indicated by a given number ; while decreas- ing it has a contrary effect. It will be necessary to bear all tliis carefully in mind. 7. Odd Numbers. One, and every succeeding alter- nate number, are termed odd ; thus, three, five, seven, &c. S. Evm Numbers. Two, and evcy succeeding alter- nate number, are said to be even ; thus, four, six, eight, &c. It is scarcely necessary to remark, that after taking away the odd numbers, all tliose which remain are even° and after taking away the even, all those which remain arc odd. We shall introduce many other definitions when treat- ing of those matters to which they felate. A clear idea of what is proposed for consideration is of tho greatest importance; this must be derived from tho definition by which it is explained. Since nothing assists both the understanding and the memory more than accurately dividing the subject of instruction, we shall take this opportunity of remarking to both teacher and pupil, that we attach much impor- tance to the divisions which in future shall actually be madr;, or shall be implied by the order in which tho different heads will be examined. b2 M I t SECTION I. ON NOTATION AND NUMERATION. 1. To avail ourselves of the properties of numbers, we must be able both to form an idea of them ouiselvcs, and to convey this idea to others by spoken and by written language ; — that is, by the voice, and by characters. The expression of number by characters, is called notation, the reading of these, numeration. Notation, therefore, and numeration, bear the same relation to each other as loriting and reading, and though often confounded, tliey are in reality perfectly distinct. 2. It is obvious that, for the purposes of Arithmetic, we require the power of designating all possible num- bers ; it is equally obvious that we cannot give a dif- ferent name or character to each, as their variety is boundless. Wo must, therefore, by some means or another, make a limited system of words and signs Buffice to express an unlimited amount of numerical quantities : — ^with what beautiful simplicity and clear- ness this is effected, we shall better understand presently. 3. Two modes of attaining such an object present themselves ; the one, that of comhining words or cha- racters already in use, to indicate new quantities ; the other, that of representing a variety of different quan- tities by a single word or character, the danger of mistake at the same time being prevented. The Romans Bimplified their system of notation by adopting the prin- ciple of combination ; but the still greater perfection of ours is due also to the expression of many numbers by the same character. 4. It will be useful, and not at all difficult, to explain to the pupil the mode by which, as we may suppose, an idea of considerable numbers was originally acquu-ed, and of which, mdeed, although unconsciously, we still avail om-selves ; we shall see, at the same time, how methods of simplifying both numeration and notation Were naturally suggested. NOTATION AND NUMERATION. 15 of numbers, im ourselves, id by written aracters. rs, is called . Notation, relation to hough often ;inct. Arithmetic, )ssible num- I give a dif- ir variety is e means or 3 and signs if numerical ^ and clear- id presently, ject present )rds or cha- mtities ; the Ferent quan- ! danger of rhe Romans ing the prin- lerfection of numbers by t, to explain suppose, an ly acquu'ed, isly, we still 3 time, how .nd notation I^.' lis suppose no system of numbors to bo as yo.i con- ^(ruered and that a licap, for oxan.ph,, of pebbles, i.s p Mcod before us that wo may discover their amount. It this IS con.sulerablo, we cnimot ascertain it l)y look- ing at tlicui all together, nor even by separatdy in- specting them; we must, therefore, have recourse t(» that contnvnuce which the mind always uses when it desires to grasp what, taken as a whole, is too great for IS powers. It we exan.inc an extensive landscape, as the eye cannot take it all in at out view, we look sue- cessively at its different portions, and form our iud.^- n.ont upon them in detail. We must act similarly with retcrerice to large nunibers ; since we cannot compre- Jiend^thcm at a single glance, we must divide them into a suHiciout number of parts, and, examining these in succession, acquire an indirect, but accuratS idea of he entire. This process becomes by habit so rapid, that It seems, it carelessly observed, but one act, thou-h It is made up of nuiny : it is indispensable, whenc^cr wo clfsire to have a clmr idea of nunibeis— which is not Iiowcver, every time they are mentioned. ' 5. Had we, then, to form for oursolVes a numerical sys eni, we would naturally divide the individuals to be reckoned into equal groups, each group consisting of S;mio number quite within the limit of our comprJlien- rum ; it the groups were few, our object would be attained vithout any further effort, .incc .ve should have acquired •urate kiiow edge of the number of groups, aiid of •n.l^er of individuals in each group, and therefore i ik.'tory, although mdu-ect estimate of the whole >.e^^ught to remark, that different persons have ery different Innits to their perfect comprehension of number; the mteliigent can conceive with ease a com- incapable of forn)mg an idea of one that is extremelv flmali. •' 6. Let us call the nmnhr of individuals that we choose cons itute a group, the rafio ; it is evident that the larger the ratio, the smaller the number of groups ano the smaller the ratio <!.- Uyo-^^ fi.p -,,-,,.i- fe^""l''^> 'in^ &ut the smaller the number of groups the be-tter. 16 NOTAT[()?J AND NUMKRATION. 7. If the groups into which wo havo divided tho objects to bo rockoueJ exceed in amount that number of whicli we havo a i)orf(!et idea, wo must continue tho process, and considorinj,' tho groups tlieniHolvos a.s indi- viduals, must form with them new groups of a higher order. Wc must thus proceed until tho number of^our highest group is sutFiciently .« ' ill. 8. Tho raiio used for groups of the second and higher orders, would naturally, but not necessarily, be the samo as that adopted for the lowest ; that is, if seven indi- viduals constitute a group of tlie first order, we would probably make seven groups of the first order constitute a group of the second also ; and so on. y. It might, and very likely would happen, that wo should not have so many objects as would exactly form a certain* number of groups of the highest order — eomo of the next lower might be left. The same might occur in forming one or moi'e of the other groups. Wo might, for example, in reckoning a heap of pebbles, have two groups of the fourth order, three of the third, none of the second, five of the first, and seven indi- viduals or "units of comparison." 10. If wo had made each of the first order of groups consist of ten pebbles, and each of the second oi-der consist of ton of the first, oacli group of the third of tm of the second, and so on with the rest, we had selected the deciiiuU system, or tliat which is not only used at present, but which was adopted by tho llcjbrew.s, Greeks, Komans, &g. It is remarkable that the language of every civilized nation gives names to the ' dilferent groups of this, but not to those of any other numerical system ; its very general diffusion, even among rude Hnd barbarous people, has most probably arisen from the habit of counting on the fingers, whidi is not altogether abandoned, even by us. 11. It was not indispensable that we sliould havo ased the same ratio for the groups of all tho diiFereut orders ; we might, for example, have made four pebbles form a group of the first order, t^velve groups of the first order a group of the second, and twen.ty grouT)a of the second a group of the third order :— iu such a i (lividcd tho ; that number b continue tho wives aH incli- is of a hii^lior lumber of our nd and higher r, be the same if seven indi- ler, we would der constitute ppcn, that wo exactly form jhcst order — ■ 10 same mip;lit groups. \Vo p of pebbles, 1 of the third, i seven indi- dor of groups second ordor e third of ton had selected only used at rews, Greeks, language of tho diiibrent ler numerical among rudo arisen from vhich is not should havo tho diiForent four pebbles •roups of the vonty o'voiiT)a :— iu such a NOTA'lloN AND NUATERATION. I7 case we had adopted a system oxacllj Hko that to \m l.i. things make a group „f the order pma' twolv-o pcnr-o a g,H,up <,f the order di/fin^s, twenty shillms^ group of tho o.lor rour.ls. While ii urns "L ad LS that the use of tho su>ne system for anplicato ■ s fV, r p^ ^:"e "';;f 'vv^'' s-utiy si,..piity ^r :;is;."ti^ ?iT •''■' H' ''''y ^'''^'-^"^ hereafter, a dance ^}t the tab OS g,v.m already, and those set down in t^^o' tt- ag of exciyjugo, will .how that a great vanoZ^X^ have actu;illy been consti-uctcd. «>> stems 12. When wo use the same ra/io for tho groups of all h.. orders, we term it a cnmuon ratio. The,^ aX' s I'-tvo boon nr> particular reason why /...slumid So Lee. seloctod as a ' connuon ratio" i„ the systeuM/f numbe' .a.,1 od, by the mode of counting on the fimrors • a„,l that It is neither so low as unnecessarily Si .Ve '0 number 01 orders of groups, nor so higlf afto oxc^^i J^o ...jceptmn of any one ibr whom the sys;;^^' . ■'"!; ^^;^y«^^^'" i'l ^^hi«li ten is the "common ntio^' r.n — ouis IS, taoroforo, a " docimal .'system" of numbe.s It the common ratio were sixty, it would b-^ a y ' . • " ^^^: :;;;'7.^"^^- i^-ony usod,;ndt^m -i e ft. 'L T P^^'^'-^ived by the tables already g.vcn toi the measurement of arcs and an-los and of time. A ^n,na,y .system wOuld have five ^ r k s' - com mon ratio - a cluoaecnal, twelve ; a vl^.i^^^l: tv.^nt^; 14. A little reflection will show that it was use-loss to give diflorent ikumos and chT-.,pfp,.« f/ "■''^^loss except to those which a "lo^'h ^ " hnt^'ILw^'l" tutos the lowest g.-oup, and to th #.:; 1 ^ilf ss.;.d^r:v^:^ '-'''''' nun;boi.iast'c:;:^;: oviaent-4i-b:u;:;;^h;;;r^t:;r^^^^^ 18 NOTATION AND NUMERATION. Nnnttt, Chitntetvt. One 1 Two 9 Til 100 .<} l''oiir 4 Kivo A Six . 6 Se'en 7 Kiiht . 8 Nu.o SI Ten 10 Hundred luo 'J'lioiisund 1,000 'J'eii tliotiKn nd" lo.wm lliiiuliud til Dusa nd ico.ooo Million . . 1,000,000 is just what wo liiivo dnno in our numnrioal py.stoni, excofit tliat we have rormcd tho nnnioH of koiuo of tlio groups )»y cmubination (»f tlioso nlrcndy used ; thus, *' tens of tliouvinTi(i.M," the grou]) n;'xt IihtIkm' th.-in lliou- sandn, is designated by a conibinntioii of words ah-cady ajtpliod to express other groups — whieli tends yet further to siniplilication. 15, ARADIC SySTEJM OF NOTATION : UdUs ol uonijiai'lson, First grotip, or units of tho second order, Seooiiii ({roup, or iiiiitti of the third ordt-.r, Tliinl Kronp, or units of tho fourth order, Fourth groii]), or units of the llfth order, l''ilih f;r()ii|), or units of the sixth order, Sixth gronii, or units of tho .seventh order, Ifi. The characters whicli express tlio nine first nuTu- bers are tho only ones used ; they are called (/i.gi/s^ JVoin tho custom of counting thoni on the lingers, already noticed — " digitus" meaning in Latin a finger ; Ihay aie also called significant figures, to distinguish them tV.,-m the cypher, or 0, which is used merely to give the digits their proper 'position with ret'crcncc to tht; dcciinal point. The pupil will distinctly remember that the place where the "• units of comparison" are to be found is that imme- diately to the left hand of this point, which, if n(»t ex- pressed, is supposed to stand to the right hand i>id(.; of all tho digits — thus, in 4(JS-7G tho 8 expresses " units of comparison," being to the left of the decimal point ; in 40 the 9 expresses " units of comparison,'"' the deci- mal point being understood to tlie riirlit of it. 17. We find by the table just given, that after the nine first numbers, the same digit is constantly repeaieil, its position with reference to tho di-ciaal point being, however, changed : — that is, to indicate each succeeding group it is moved, by means of a cyplier, one place farther to the loft. Any uf the dibits may be uaed to N. fiorionl pystoni, of 8omo of (lio y usod ; thus, rlier tlinn tliou- wotds nlrcjuiy !U(Is yot further Chantetvt. 1 a 4 A 6 7 8 SI 10 10(1 I,(J()() 10,(HI0 NOTATION AND Nt/MERATIOrf 19 !<1 ind niKnnd d thousand l(:0,(il)(i l,()00,f)(;o nine fii-Ht miTii- iod ili.gih^ JVoiii iiigor.s, already iigcr ; Ihay are ish thciiu fr.,"in give tlie di^i<s ! decimal p. int. he place where d \H that imtue- ieh, if nut ex- t hand side of presses " uuitH decimal point ; son,'' the deci- ■ it. that after tlse antly repeated, d i)oint being, ach succeeding' lier, one place ay be used to flxpreHs its respectiv(3 n.iniber „f any of the .rroups •-. thus S would be eight u ...j,, ./.nnpari^onTH^ oi-ht groups ot tho first ord<n-, or ei-dit " tens" of Mmpic units ; 800, eight groups of tho .second, or' unit^ f tho h.rd order ; and so on. Wo might use any of tho digits With tho different groups ; thus, for examDio 7 ;;;; r^'n^^v'^'^ "^i"^^ 3?o;those'of;he " S; •Z'to r *'^:: ^^^1^''" •?^'*^ J™ i'l f>ill would be 5000, 3 0, 70 8, or for brevity sake, 5378-for wo never nm tho cypher when we can supply its place by a si^^nificant ftguro, and it is evident that in 5378 t!ie 378 ko p^^^^^^^ T) tour p.aees from th. decimal point (understood), fult aa wo as cyphers would have done ; a\so the 78 k^ee'pT ho 3 Y the third, and the 8 keeps the 7 in tho second placo. lb. It IS important to remember that each di^rit hag two values, au absoUta and a relative; tho absolute un " m v' h ""'"i '' ""^^^^ '^^'^'^'^^ whatever tte units may bo, and is unchangeable: thus 6 alwavs I'/oans SIX, sometimes, indeed, Sx tens, at otlu^r tSa Hix hundred, &c. The relative value depends on the o dcr of units mdicated, and on the nature if tho "unit ot comparison." * _ la. What has been said on this very important snli- l;;ct, IS intended pi-incipally for the teLheir both t o.-dmary amoun of industry and intelligence wil be a ci ld,_ paitieulariy it each point is illustrated by an appropriate example ; the pupil may be made f\ f in .stance, to arrange a numbe/of pebbles in groups son "- ti.nes of one, sometimes of another, and soiZinies of S'od f 1 ;n7 • 1 1 '°'"P^'''^'' ^^'^"^o^ occasionally ,, Changed tiom individuals, suppose to tens, or hundreds op \ scores, or dozens, &c. Indeed the pupils J fbe well ; oquamted witli these introductory ma^tters otherw^o ., verj acnnite ideas of many th n'>-s thev will be poIl/i inijc£ii.y to understand. A.n" t"0'ihia Un •f-.ixr i i ,^ toaoho. at thi. „.,„,, ,,ni f,o we,f;o,«i;n;'ttt (i ■ 111 20 NOTATION AND NrMliRA'I'IOV. and rapidUj^ with Aviiicli tlic sdiolar will cafterwardg adviincjo ; to be assured of this, ho has only to recol- lect that most of his future reasonings will be derived from, and his explanations grounded "on the very prin- ciples we have endeavoured to unfold. It may be taken as an important truth, that what a child learns without undcratanding, he will acquire with disgust, and will uoon cease to remember ; for it is with children as with persons of more advanced years, when wo appeal suc- cessfully to their understanding, the pride and pleasure they feel hi the attainment of knowledge, cause the labour and the weariness which it costs to be under- valued, or forgotten. 20. Pebbles will answer well for examples ; indeed, their use in computing has given rise to the term calcu- lation., " calculus" being, in Latin, a pebble : but while the teacher illustrates what he says by groups of par- ticular objects, he must take care to notice that hi^^ remarks would be equally true of any others. He must also point out the difference between a group and its e(juivak'nt unit, which, from their perfect equality, are generally confounded. Thus ho may show, that a penny, while C(iual to, is not identical with four farthings. This seemingly unimportant remark will be better appre- ciated hereafter ; at the same time, without inaccuracy of result, we may, if we please, consider any group dther as a unit of the order to wliieh it 1)clongs, or so many of the next lower as are equivalent. 21. lloman Notaiion. — Our ordinary numerical cha- racters have not been always, nor every where used tc express numbers ; the letters of the alphabet naturally pi-esonted themselves for the pui-pose, as being already familiar, and, accordingly, were very generally adopted— for example, by the Hebrews, Greeks, llomans, &c., each, of course, using their own alphabet. The pupil should be acquainted with the lloman notation on account of its beautiful simplicity, and its being still employed in inscriptions, &c. : it is found in the follow- ing table : — ill afterwardg only to rccol' dll be derived :lio very prin- may bo taken learns without ;nst, and will ildren as with e appeal suc- and pleasure 50, cause the to be under- ples ; indeed, le term calmt- le : but while roups of par- tice that \m rs. He must ;roup and its equality, are that a penny, 'things. This better appre- Lit inaccuracy r any group )elongs, or so imerical cha- vhere used tc .bet naturally being already lly adojited — ■ iomans, &c., u The pupil notation on ts being still in the follow- NOTATIOi'T AND NUMERATION. ROMAN NOTATION. 31 Characters. I. n. in. . Anticipated change IIII, or IV. OJiauge . . V. . VI. . VII. . VIII. . Auticipaletl change IX. Change X. XI. XII. XIII. xtv. XV. XVI. XVII. XVIII xrx. XX. A uticipated change X [, Change . . L. . . T'X., &c. Anticipated change XO. Change . . C. , . ". CO., &c. Anticipated change CD. Cliango . . D. orr>. Anticipated change CM Ciiauge , ' . U. or CIq y. or 1,30 &c JSiimhers Exprrsstd. . One. . Two. . TJiree . Four. . Five. . Six. . Seven. . Eight. . Nine. . Ten. . Eleven. . Twelve. . Thirteen. . Fourteen. Fifteen. Sixteen. Seventeen. Eighteen. Nineteen. Twenty. Thirty, &c. Forty. Fifty. Sixty, &c. Ninety. One hundred. Two hundred, kc Four hundred. Five hundred, &c Nine hundred. One thousand, &c. Five thousand, &c. X. or CCIoo • Ten thousand, &o. laop. . Fifty thousand, &cj. '^'-'CiOOO. • One liundred thous.and, &c . 2,2. Thus we find that the liomuns used vnrv fe^7 eiiaracters-fower, iridoed, than we do, althou-Wl our sjstoMi ,s stdl more snaplo and eiToctive, from our ap,>K-- ing.tli. prmc.pl. of "position," unkuo-.vn to them ^ llicy expressed all numbers by the followiiv. symbol. or combmatiou.3 of them : I. Y. X. L. (J. D. ov U T" orCLo.^ In cojistrueting their system, they evidently had a (pmary m view ; that is, as we have ^rid, an. I wlucii nve would be the common rallo ; for we find that luey ciians'tiu th heir eharactur, not only at ten, toij t iine» i i in. 22 WOTAT ^,^ Aww wwMEKATION. ten, &c., but also at five, ten times five, &c. : — a purely decimal system would suggest a change only at ten, ten times ten, &c. ; a purely quinary, only at five, five times five, &c. As far as notation was concerned, what they adopted was neither a decimal nor a quinary system, nor even a combination of both ; they appear to have supposed two primary groups, one of five, the other of ten " units of comparison ; " and to have formed all the other groups from these, by using ten as the common ratio of each resulting series. 23. They anticipated a change of character; one unit before it would naturally occur — that is, not one " unit of comparison," but one of the units under consi- deration. In this point of view, four is one unit before five ; forty, one unit before fifty— tens being now the units under consideration ; four hundred, one unit before five hundred— hundi-eds having become the units con- templated. 24. When a lower character is placed before a higher its value is to be subtracted from, when placed after it, to be added to the value of the higher ; thus, IV. means one less than five, or four ; VI., one more than five, or six. 2b. To express a number by the Roman method of notation : — Rule.— Find the- highest number within the given one, that is expressed by a single character, or the " anticipation " of one [21] ; set down that character, or anticipation— as the case may be, and take its value from the given number. Find what highest number less than the remainder is expressed by a single charac- ter, or " anticipation ; " put that character or " anticipa- • tion "^ to the right hand of what is already wi'itten, and take its value from the last remainder :' proceed thus until nothing is left. Example.— Set down the present year, eighteen h'mdred and forty-four, in Roman characters. One thousand, ex- pressed by M., is the highest number within the giynn one, indicated by one character, or by an anticipation; we pu k down and take one thousand from the given number, which loaves ^S r. ic. : — a purely tily at ten, ten EJve, five times ed, what they linary system, ppear to have , the other of formed all the LS the common laracter ; one it is, not one 3 under consi- ne unit before eing now the ne unit before he units con- jed before a when placed ligher ; thus, I., one more n method of lin the given icter, or the at character, take its value ^hest number lingle charac- or " anticipa- • written, and proceed thus teen h'lndred thousap.d, ex- the givon one, ; we pu k down which loaves NOTATIOM AND NCTJIERATION. 23 'IS' eight hundred and forty-four. Five hundro „ tho highest number within tho last remainder (e /./^ fu.r,dred and iory-four) expressed by one character, or jin "antici- pation ;■' we set down D to the rin;ht liand of M JNID, and take its value from eight hundred and forty-four, which leaves three hundred and forty-four. In this^the hio-host number expressed by a single character, or an "anticipa- tion, ' IS one hundred, indicated by C ; which we set down ; and tor the same reason two other Cs Tw , MDCCC. This leaves only forty-four, tho highest number withic which, expressed by a single character, or an "anticipation '' IS torty, AL— an anticipation ; we set this down alsof _ MDCCCXL. Four expressed by IV., still remains; which, being als« added, the whole is as follows:— « uj, ms* MDCCCXLTV. 26 Posiiion.—The samp .jharacter may have dif- terent values, according to tie place it holds with refer- ence to the decimal point, or, perhaps, more strictly, osiiion comparison." This is the principle of 27. The places occupied by the units of the different orders, according to the Arabic, or ordinary notation [lo] , may be described as follows :-units of comparison, one place to the left of the decimal point, expressed 01- understood ; tens, two places ; hundreds, three places tl K 1 F""^'^ '^'''"^•^ ^' "^^'^^ '^'^ ^^™ili^r with these as to be able, at once, to name the « place" of any order of units, or the " units" of any place ^ 28 When therefore, we are desh-ed to write anv number, we have merely to put down the digits expres- sing the amounts of the different units in their prler places, according to the order to which each belongs, it, in the given number, there is any order of which domi u the place belonging to it ; the object of which 2r^ ^7 'I::;, "Snifica. t figures in^ thpir own posi- mns. A cypher produces no effect wh between significant figu^^s and the 0536, 636-0, and 536 on it is not decimal point; thus would mean the same thing — the 24 NOTATION AND NUMERATION. second s, however, the correct form. 536 and 5360 dre different ; in the latter case the cypher affects the value because it alters the position of the digits. ' Example —Let it be re uired to set down si^ hundred h' ; ' Sr "^*^,"«' ^^2 : without the cypher, the six would 29. In numerating, we begin with the digits of the highest order and proceed downwards, stating the num- ber which belongs to each order, fe nuiu clJhV'tf'^''^'' notation and numeration, it is usual to divide the places occupied by the different ordei of units into periods ; for a certain distance the Englisn and 1 rench methods of division agree ; the English bHlion IS, however, a thousand times greater than the French Ihis discrepancy is not of much importance, since we aie rarely obkged to use so high a number,-we shall prefer the IWh method. tS give some idea of the amount of a billion, it is only necessary to remark that according to the English Method ot' notation 'there lias not been one billion of seconds sin e the birth of thrist. Indeed, to reckon even a million, counting on an average^ three per second for eight hours a day, would require nearly 12 days. The following are the two methods. ° \* Jill ENGLISH METHOD. Twin nnn- B^^ions. Millions. 000-000 000-000 000-000 Units. 000-000 Billions. Quiidivila. 'J'ens. Unita. FRENCH METHOD. Millions. Thousands. Units. Himd. .ens. Units. H„nd. Ten.. Unita. Hnnd, Ton.. Unit* 000 000 000 so Use of Periods.— Let it be required to read off the following number, 576934. We put the first point to the left of the hundreds' place, and find that there are exactly two periods— 576,934 ; this does not always occur, as the higlicst period is often imperfect, consisting oniy 01 one or two digits. Dividing "the number thus NOTATION AND NUMERATION. and 5360 jlre cts the value, 1 six hundred the two in the ^her between the six wouhi mean not six digits of the ng the num- t is usual to it ordei of Englisii and iglish billion the French, ee, since we p, — ^we shall idea of the •emark, that ation, there he birth of counting on ours a (lay, ing are the Units. 000-000 Units. nnd. Tpi)«. Unit* to read off i first point it there are not always , consisting imber thus 2b into part,^, shows at once that 5 is in the third place of the second period, and of course in the sixth place to the 4eft hand of the decimal point (understood) ; and, therefore, that it expresses hundreds of thousands. The 7 being in the fifch place, indicates tons of tliousands ; the tJ in the fourth, thousands ; the 9 in the third, hun- dreds ; the 3 in the second, tens ; and the 4 in the first, units (of " comparison ") . The whole, therefore, is five hundreds of thousands, seven tens of thousands, six thousands, nine hundreds, three tens, and four units,— or more briefly, five hundred and seventy-six thousand, nine hundred and thirty-four, 31. To prevent the separating point, or that which divides into periods, from being mistaken for the decimal point, the former should be a comma (,)— the latter a tuli stop (•) Without this distinction, two numbers Y.o^'i^f'' ^?^ different might be confounded : thus, 498- /63 and 498,763,-one of which is a thousand times greater than the other. After a while, we may dispense witli the separating point, thouo-h it is conve- nient to use It with considerable number, as they are tiien read with greater case. ♦ 32. It will _ facilitate the reading of large numbers not separated mto periods, if we begin with the units of comparison, and proceed onwards to the left, saymg at the fi,« digit ;' units," at the second " tens," at the tlnrd 'hundreds," &c., marking in our mind the deno- mination of i\iQ highest digit, or that at which we stop. We then commence with the highest, express its number and denommation, and proceed in the same way with each, until we come to the last to the right hand. ■ .+ !;^'-'^^^P;'J=— J^et it be required to read off 6402. Lookins ti t;^-'tlT"'^VV'^^r ^^y """^*^' ' '^' theO, ''tens;i ?,fv^ fi ' V^^"^,^*«.<i«i" .a^d at tlie C, -thousands.- The .it.„er, therefore, being six thousands, the next di-it is f(,nr hundrod., &o Consequently, six thousands, fourlunu r ds lV^ rf two units; or, briefly, six tliousand four hun drcd and two, is the reading of tlie given number. "m" ^^'t^ "'''^ ^^° ^•^^'^ *'^ facilitate notation, 'i^he pupil will fu-st write domi a number of Doiit^ds of cyphers 26 NOTATION AND NUMERATION. to roprosenf, tlio places to be occupied by the varioua orders of imit.s. Jle will then put the digits express- ing the diflerent denominations of the given number under, or instead of those cyphers which are in corres- ponding positions, with reference to the decimal points bogmumg with the highest. ExAMPLE.-Write down three t'nousand rlx hundred and btty-tour The highest dent ■< • -^a being thousands, will occupy the fourth place to thr ' ' the deoimal point. It will be enough, therefore, to ^r down four cyphers, and under them the corresponding digits-that expressing the thousands under the fourth cypher, the himdreis under tho third, the tens under tho second, and the units under the hrst; thus 0,000 3,654 A cypher is to be placed under any denomination iu which there is no significant figure. Example.— Set do^vn five hundred and seven thousand, and sixty-three. ' 000,000 507,063 After a little practice the periods of cyphers will become unnecessary, and the number may be rapidly put down at once. 34. The units of comparison are, as we have said, always found in the first place to the left of the decimal point ; the digits to thr left hand progressively increase in a tenfold degree—those occupying the first place to the left of the units of comparison being tea times greater than the units of comparison ; those occu- pying the second place, ten times greater than those which occupy the first, and one hundred times greater than tho units of comparison themselves ; and so on. Moving a digit one place to the left multiplies it by ten, that is, makes it ten times greater ; moving it two places multiplies it by one hundred, or makes it one hundred times greater ; and sc of the res^ If all the digits of a quantity be moved one, two, &c., places to the left, the whole is increased ten, one hundred, &c., times— as tho case may be. On the other hand, moving i NOTATION AND NUMERATION. 87 ' the various '^its express- fen number, re iu corres- uial point-— hundred and ousands, will lal point. It 3yphora, and f pressing the ids under tho ts under the mination ia sn thousand, yphers will be rapidly have said, eft of the 'ogressively ig the first . being ten those occu- than those ics greater md so on. plies it by i^ing it two kes it one If all the , places to idred, &c., ad, moving a digit, or a quantity one place to the right, divides ifc. by ten, that is, makes it ten tim^-s smaller than before ; moving it two places, divides it by one hundred, or niiikes it one hundred times smaller, &c. So. We possess this power of easily increasing, or diminishing any number in a tenfold, &c. degree, whetlicr the digits are all at the right, or all at the loft of the docimal point ; or partly at the right, and partly at the loft. Though we have not hitherto considered quautitiea to tho left of the decimal point, their relative value will be very easily understood from what wo have already said. For the pupil is now aware that in the decimal system the quantities increase in a tenfold dej^ree to the loft, and decrease iu the same degree to "the right ; but there is nothing to prevent this decrease to "the right from proceeding beyond the units of comparison, and tho decimal point ;— on the contrary, fi-oni the very nature of notation, we ought to put quantities ten times loss than units of comparison one pl-ice to the right of them, just as we put those which are ten times less than hundreds, &c., one place to the right of hundreds, &c We accordingly do this, and so continue the notation not only upwards, but downwards, calling quantities U the left of the decimal point integers, because none of them is loss than a whole " unit of comparison :" an^ those to the right of it decimals. When there are deci- mals m a given number, the decimal point is actuallv expressed, and is always found at the ri.<(ht hand side ot the units of comparison. 30. The quantities equally distant from tho unit of comparison bear a very close relation to each other which IS indicated even by the similarity of their names • those which are one place to the left of the units of com- parison are called " tens," being each identical with -«r equivalent to ten units of comparison; those which are one place to the riffht of the units of comparison tim called tenths," each being the tenth part of, tlint is, ten times as small as a unit of comparison ; quantities two places to the Ipfi of the units of comparison are called imndrcds" being one hundred -times greater ; and those two places to the rigAl, " hundredths," beincr one S8 h )••; ■M NOTATION AND NUMERATION hun red times loss t u.n tlio units of comparison ; and .a ot .11 11.0 o hers to tl.c right and loft. This will bo mo o evident on in.sp.,cting the following tabic :-- Asrcirling Scries, or Integers Uiic Liiit Ilumlred . . jqq Thousand . . 3,000 Ion tliou,s!ui(.l . 10,000 Hundred tlioiisand 100^000 &c. II' IVsccmlin-.' Sories, or DccliiMls. Ouo Unit. Tenth. Htindrcdtli. Tiioijsiindth. Ten-tliousfindth. Hundred- thousandth. &G. •1 •01 ■001, •000,1 •000,01 We have seen that when we divide integers into periods Ifih.^ .separating point must be put to the rid.t of the thousands; m dividing decimals; the first poin nius^ be put to the right of the thousandths. ^ 37. Oare must be taken not to confound what we Tl "ul !^''^'^': ^v'itli what wo shall hereafter des I! not irlent ica lly the same quantities-the decimals beino- wha sha 1 bo termed the " quotients" of the con-e.^ pondmg decimal ft.actious. This remark is made hero ^ i^n''? %T'"''''"'^^" ''^'^ ^^^ ^^' subject, in those who already know something of Arithmetic T'x. '^^"^l^ ^'^ ^'^ ^■'^''^"^^ ^"^'' treating integers and deci- mals bj dfent rules, and at ditForc;^t tin!es, since the^ follow precisely the sa.ne laws, and constitute parts of Uie very same series of numbera ]]esides, any quantity nay, as far as the decimal point is concerned, be ex- pressod m diflerent ways; tbr this purpose ^e ha;e Jierely to change the unit of comparison. Thus let it be required to set down a number indicating five hun- dred and seventy-four men. If the " unit of compari- son ho one man, the quantity would stand as follows, &74. If a band of ten men, it would become 57-4— f^r as each man would then constitute only the tenth pa,J of the "unit of comparison," four men would be only our-tenths, or 0-4 ; and, since ten men would forn. bu^ one unit, seventy men would be merely seven uni^s of comparison, or 7 ; &c. Again, if it wen. a band of one ImM mm, the number must be ^vritten 5-74 ; an-] I'lbiij, ir a miiKi ol a Inuusand 9mij it would be 0-574 m NOTATION AND NUMERATION. 29 son ; and so will be more , or DooJiiwls. :h. Ith. siindth. •tliousfindth. Into periods to the rJLdit first point d what we after dosici;- equal, but inials beiniT the corres- ;ide here to t, in those and deci- sinco they 3 parts of y (juantity ed, be ex- ' wo have 'hus, let it 'l five him- conipari- ts follows, 37-4— for, enth pact 1 bo only form but I units (»f lid of one ■74 ; and be 0-574 Should the " unit " bo a band of a dozen, or a score men, the change would be still more complicated ; as, not only the position of the dechual point, but the very digits also, would be altered. 39. It is not necessary to remark, that moving the decimal point so many places to the left, or the digits an equal number of plaees to the right, amount to the same thing. Sometimes, in changing the decimal point, one or more cyphers are to be added ; thus, when we move 42 '(3 three places to the left, it becomes 42600 ; when wo move 27 five places to the right, it is '00027, &c. 40. It follows, from what we have said, that a deci- mal, though less than what constitutes the unit of com- parison, may itself consist of not only one, but several individuals. Of course it will often be necessary to indi- cate the " unit of comparison,"— as 3 scores, 5 dozen, 6 men, 7 companies, 8 regiments, &c. ; but its nature does not affect the abstract properties of numbers ; for it is true to say that seven and five, when added together, make twelve, whatever the unit of comparison may be :— provided, however, that the sa7?ie standard be applied to both ; thus 7 men and 5 men are 12 men ; but 7 men and 5 horses are neither 12 men nor 12 horses.; 7 men and 5 dozen men are neither 12 men nor 12 dozen men. When, therefore, numbers are compared, &c., they must have the same unit of comparison, or — without alterin^r their value — they must be reduced to those which have"? Thus we may consider 5 lens of men to become 50 individual men— the unit of comparison being altered from ten men to om man, without the value of tho quantity being changed. This principle must be kept m mind from the very commencement, but its utility will become more obvious hereafter. EXAMPLES IN NUMERATION AND NOTATION. JVoiaiion. 1. Put down one hundred and four 2. One tliousand two hundred and forty 3. Twenty thousand, three hundred and forty-five ^iu.i. 104 1,240 20,345 ^ i ! mV. : t i' i •so NOTATION AND NUMERATION. 5. G. 7. Two Imndrod and tliirty-fuur thousand, fivo hundred and sixty-seven Three hundred and twenty-nirio tliousand, seven hundred and sevcnly-nine Seven hun(h-ed and nine tliousand, eight hun- dred and twelve . . *! . Twelve hundred and fo/ty-soveu tliousand, four hundred and tiCty-sovon 8. One million, three hundred and ninety-seven thousand, four hundred and seventy-live 0. Put down fifty-four, seven-tenths 10. Ninety-one, fivo hundredths . 11. Two, three-tenths, four thousandths, and four hundred-thousandths 12. Nino thousandths, and three hund]-ed thou- sandths • • . . . 13. Make 437 ten thousand times greater 14. Mako 2 7 one hundred times greater 15. Mako 0056 ten times greater . . 10. Make 430 ten times less 17. Mako 2-75 one thousand times les^i . Jln.i. 234,507 320,771) 709,812 1,217,457 1,397,475 54-7 • 0105 2*30401 000903 4,370,000 270 0-56 43 000275 Numeration 7. read 8540320 5210007 Gi.)30405 50- 0075 3' 000000 00040007 2. — 407 8. 3. — 2700 • 0. 4. — 5000 10 6. — 37054 n. G. — 8700002 12. 13. Smnd travels at the rate of ahout 1142 feet in a seeond ; light moves ahout 195,000 miles in the same time. 14. The sun is estimated to be 880,149 miles in diameter: its size is 1 377,013 times greater than that of the earth. 15. Tho diameter of the planet mcreurv is 3,108 miles, and his distance from the sun 30,^; 14,721 miles. 10. The diameter of Venus is 7,498 miles, and her dis- tance from the sun 08,791,752 miles. _17. The diameter of the earth is a])out 7,904 miles: it is 95,000,000 miles from tlie sun. and travels round the latter at the rate of upwards of 08,000 miles an hour. 18. The diameter of the moon is 2,144 miles, and her dis- tance from the earth 230,847 )niles. 10. The diameter- of Mars is 4,218 miles, and his distance from the sun 144,907,030 miles. 20. The diameter of Jupiter is 89,009 miles, and his dis- tance from tho sun 494,499,108 m.ilcs. I flvO Jtnt. . 234,507 and, . 320,771) liiiii- . 709,813 ami, . 1,217,457 3vcn . 1,397,475 54-7 • 0105 four . 2.30401 tiou- . 000903 . 4,370,000 270 0-50 43 . 000275 W32C) 10007 M)405 0075 oooon 040007 ^42 feet in a e same time, i in diameter j the earth. ! 3,108 miles, I. and her dis- 1 miles : it in md the latter , and her dis^ I his distance , and his dis- NOTATION AND NUAIKRATION. 31 ) 21. The diameter of Saturn is 78,730 miles, and hia die- taiico from the sun 907,089,032 miles. 22. 'J 'ho length of a pendidmu which would vibrato Hcconds at the ci^uator, is 39011,084 inches; in the latitude of 4o degrees, it is 39116,820 inches; and in the latitude of 90 degrees, 39-221,95G inches. 23. It has been calculated that the distance from the earth to the nearest fixed star is 40,000 times the diameter of the earth's orbit, or annual path in the heavens ; that is, about 7,000,000,000,000 miles Now suppose a camion ball to fly from the earth to this star, with a uniform velocity equal to that with which it first leaves the mouth of the gun~-say 2,500 feet in a second— it would take nearly 1,000 years to reach its destination. 24 A p.iece of gold equal in bulk to an ounce of water, would weigh 19258 ounces; a piece of iron of exactly the sanie size, 7788 ounces; of copper, 8788 ounces; of lead. 11'352 ounces; and of silver, 10474 ounces. NoTK.— The examples in notation may be made to answer for numeration ; and the reverse. QUESTIONS IN NOTATION AND NUMERATION. [The references at the end of the questions show in what paragraphs of the preceding section the respective answers are principally to be found.] 1. What is notation } [1]. 2. What is numeration .? [1]. 3. How are we able to express an infinite iriety of numbers by a few names and characters ? [31 . 4. How may we suppose ideas of numbers to have been origmally acquired > [4, &c.]. ^5. What is meant by the common ratio of a system of numbers .> [12] . j ^^ 6 Is any particular number better adapted than another for the common ratio .? [12]. ■ ratio .?^[11]^^''' '^'*'''^' ""^ numbers without a common 8. What is meant by quinary, decimal, duodecimal, vigesimal, and sexagesimal systems ? [13]. 9. Explain the Arabic system of notation ? ri5"I 10. What are digits .? [161 ■^' 1 1 TT *l 1 . 1 . -^ow arc tliey maac to express all numberg ? [17] . I ■^^ ti S9 NOTATION AND NUMERATION. ■*or of units of a lower order precisely the same thing ? 14. Have the characters wo use, always and every Inhere been cmp oyed to express numbers ? [211 ^ rh.!i-ff ♦ ''i^' ^r^"^^^ P^>""*' «^»d the posit on of figiJesfe]."' ''^ '^ °^'"« affect- significant F^n'T,^''?^ !? the difference between the English and aolLIns^l '^^ '*^"^° '^^^^^'^ -*^g- -^ «.i^; T^'''* \' "'"''''* H *^^ ascending and descending X.^ [36] ' '" "^ "'" the/ related to each n„fJ' f '''V*^^* i". expressing the same quantity, we mist place the decimal point differently, according to the unit of comparison we adopt ? [38] 22. What effect is produced on a digit, or a quantity by removing it a number of placs to the right, or left or similarly removing the decimal point ? [34 iid 39] ^f. le and relativo 33 luivalent num- le samo thiujr ? ays and every ' [211. :on? [22, &c.]. lie position of ICG to it ? [26 3ct significant e English and )eriod8? [29 J. integers and nd descending slated to each quantity, we according to or a quantity right, or left, [34 and 39] SECTION 11. THE SIMPLE RULES. SlftlPLE AUDlTiON. oc,'; tLv""t'"ir "•'""«"'' J'y ""7 arithmotioal pro- ocas, they are cither morcasod or (iimiDislii.r) • if \„ f?™toe-££SS:?- but vo may have m mmm of tl,™ , ""^ """^' called "Multiplieator''\?l Zv'' ™ ^l'""'/'' '" .««, but Jheir'n„„.b?r'i. imlS^t' Ty™ I^^Jf Z quantities to be Uod-L^lnlL'T' "" 1";'"'"='' "^ *« til" kind l,„t ..„r.i ' '""'''Pl'ca'ion restricts us as to .•eally comprehended under trtl' ™^^SS' "" £.|s-^rn^i:-t^-r=: ^..o%fu^';^iftiret;f.i,;L7;r^^^ means, that G is to be S d'od "o 8 wf "^ ^ ^ "" ?' prcfacd, the positive is undtstood '"" "" "^^ ^ to 16. ' ^^'^ ^'^^ ''^^" "^ 9 i^^^ 7 is equal II, ^if- 34 ADDITION. Quantities connected by the sign of addition, or that of equality, may be read in any order ; thus if 7 + 9=16, it is true, also, that 9 + 7=16, and that 16=7+9, or 9+7. 5. Sometimes a single horizontal line, called a viii- mlum, from the Latin word signifying a bond or tie, is placed over several numbers ; and shows that all the quantities under it are to be considered, and treated as \)\xt one ; thus in 4+7=11, 4 + 7 is supposed to form but a single term. However, a vinculum is of little consequence in addition, since putting it over, or remov- ing it from an additive quantity — that is, one which has the sign of addition prefixed, or understood — does not in any way alter its value. Sometimes a parenthesis ( ) is used in place of the vinculum; thus 5+6 and (5+6) mean the same thing. 6. The pupil should be made perfectly/ familiar with these symbols, and others which we shall introduce as we proceed ; or, so far from being, as they ought, a great advantage, they will serve only to embarrass him. There can be no doubt that the expression of quantities by characters, and not by words written in full, tends to brevity and clearness ; the same is equally true of the processes which are to be performed — the more con- cisely they are indicated the better. 7. Arithmetical rules are, naturally, divided into two parts ; the one relates to the setting down of the quan- tities, the other to the operations to be described. We shall generally distinguish these by a line. To add Numbers. RirtE. — T. Set down the addends under each other, so that digits of the same order may stand in the same vertical celumn — units, for instance, under units, tens under tons, &c. II. Draw a line to separate the addends from their mm. III. Add the units of the same denomination together, boci:!nninf» at the rijzlit linnd side. IV. 9 the sum of any column bo less than ten, set it down under that column ; but if it be greater, for every ADDITION. 35 s from their ten it contains, carry one to tlie next column, and Dut down only what remains after deducting the'tens-^tf 'wthing remains, put down a cypher ' V. Set down the sum of the last column in full. 8. Example.— Find the sum of 542-|-375-^984— 375 } addends. 984 J 1901 sum. Tt j'c . ' '^' ^"*^ ^' which are "hundreds" in nnnthoiT JO £t^pr:s^-i,--/-|SS Vn5? ^'""Pf' ^'ly"'" ^^^™^^ notation, can easily find l7d"Krof " '-: ^" ' ^^^"" ^^-^^r 5 sini all tiic dio ts that express it, except one to the ri^vht hnnrl j^ide, ml indicate the number of '4ens'' it JonVt „s thus m 14 there are 1 ten, and 4 units • in li^ S J ' -d 2 units ; in 143, 14 tens, and's unit's, L'' ' *"'' The ten obtained from the sum of the units alon<r wUh « :;reds._„„d write down a cypher in theTns. pLe" rf S^^ The two hundreds to be "carried " idflorl +a o Q i k As there are no thousands in the next cnlumn ^h.i- • ti.o last clnm i^'full. ""' ™*' ™ '<'' •^"™ "'« »"■» «f «lS-ri^Un<SLSimr^;rlL-! !i;!i 36 ADDITION. that we may easily find those quantities which are to be added together ; and that the value of each digit may be more clear from its being of the same denomination as those which are under, and over it. Reason of II.— We use the separating line to prevent the sum from being mistaken for an addend. Reason of III.— We obtain a correct result only by adding units of the same denomination together [Sec. I. 40] :— hun- dreds, for instance, added to tens, would give neither hnndreda nor tens as their sum. We begin at the right hand side to avoid the necessity of more than one addition; for, beginning at the left, the process would be as follows — 542 375 984 1,700 190 11 1,000 800 100 1 1,901 The first column to the left produces, by addition, 17 hun- dred or 1 thousand and 7 hundred ; the next column 19 tens, or 1 hundred and 9 tens ; and the next 11 units, or 1 ten and 1 unit. But these quantities are still to be added :— beginning again, therefore, at the left hand side, we obtain 1000, 800 100° and 1, as the respective sums. These being added, give 1,901 as the total sum. Beginning at the right hand rendered tho successive additions unnecessary. • ^\f^T'^^ OF IV.— Our object is to obtain the sum, expressed m the highest orders, since these, only, enable us to represent any quantity with the lowest numbers ; we therefore consider ten of one denomination as a unit of the next, and add it to those of the next which we already have After taking the " tens » from the sums of the different columns, we must set down the remainders, since they are parts oiihQ entire sum; and they are to be put under the CO umns tliat i.roduced them, since they have not ceased to Dclong to the denominations in these columns Reason or V.-It follows, that the sum of the last column ot„ \ ff * ^""y""^ '!" ^''^^ ' ^'^^' ('" *he above example, for in- it contains ""'' '" "°*^^"g *o be added to the tens (of hundreds) 10. Proof of^Addition.—Oxxi off the upper addend, by a separating line ; and add tlie sum of tho (|uantitie» ADDITIOIf 87 re to be added be more clear ose which are ;o prevent the mly by adding I. 40] :-~hun- ther hundreds e necessity of ft, the process tion, 17 hun- iumn 19 tens, , or 1 ten and I : — beginning 000, 800, 100, id, give 1,901 rendered the im, expressed to represent fore consider und add it to the diflferent Qce they are lit under the lot ceased to 3 last column ™ple, for in- 9f liundreda) lor addend, (|uantitie» under, to what is above this line. If all the additions have been correctly performed, the latter sum will be equal to the result obtained by the rule : thus— 5,673 4,632 8,697 2,543 21,545 sum of all the addends. 15,872 sum of all the addends, but one. 5,673 upper addend. 21,545 same as sum to be proved. This mode of proof depends on «ie fact that the whole in equal to the sum of its parts, in whatever order they are Sf i' • Ji'*i' ^'""^l^. ^° *^® objection, that any error com- nutted in the first addition, is not unlikely to be repeated in the second, and the two errors would then conceal each other To prove addition, therefore, it is better to go through the process again, beginning at the top, and proceedinc downwards. From the princ^le on whicS the JtS of proof IS founded, the result of both additions-the direct and reversed— ought to be the same. It should be remembered that these, and other proofs of shfcf U ?, nnl '• ^^''"'^ '"'''^? "" ^'^^ d«g^«« «f pro'babiUty, since It is not in any case quite certain, that two errors cal- culated to conceal each other, have not been committed. ul'i^'^^^^^ Qwaw^2Vw!5 containing Decimals. —From. What has been said on the subject of notation (Sec. I. db) It appears that decimals, or quantities to the right hand side of the decimal point, are merely the continu- ation, doivnwards, of a series of numbers, aU of which to low the same laws ; and that the decimal point is mtendpd not to show that there is a difference in the nature of quantities at opposite sides of it, but to mark ni f- !i^v ""'^ 1 «TP""^°^" ^ Pl^^^d. Hence the mJo tor addition already given, r.pplies at whatever side a I, or any of the digits in the addends may be found it IS necea-ary to remember that the decimal point in the sum should stand precisely under the decunal points of the ^a.lends ; smce the digits of the sum must beffrom the very nature of tl^- — p°"=, rm a uT,oui.,num ,, . ■' -''--^-'; '•' ^!-'- inuce.^3 [D , ot exactly the same value, rospectirely, as the digits of the addends under g2 38 ADDITION. whicli thoy are ; antl if set down as tlioy should bo, their denominations are ascertained, not only by their position with reference to their o i decimal point, but also by their position with reference to the digits of the addends above them. Example. 263-785 460-602 637 -008 626-3 1887-695 It is not necessary to fill up the columns, by adding cyphers to the last addend ; for it is sufficiently plain t^at we are not to notice any of its digits, until we come to the third column. 12. It follows from the nature of notation [Sec. I. 40], that however we may alter the decimal points of the addends— provided they are all in the same vertical column— the digits of the sum will continue unchanged ; mus in the followin<y : — 4785 8257 6546 14588 478-5 325-7 654-6 1458-8 47-85 32-57 66-46 145-88 •4785 •3257 •6546 1^4588 •004785 •003257 •006546 •014588 I: EXERCISES. (Add the following numbers.) Addition, Multiplic (1) (2) (3) (4) (5) 4 8 3 6 4 6 4 9 6 4 3 7 7 6 4 6 6 6 6 4 7 2 5 6« 4 ~~ — — — — _ — — — — (10) (11) (12) 6763 3707 2867 2341 2465 8246 5279 5678 1239 (6) 9 9 CO 9 9 9 (13) 6978 3767 1236 Involution, (8) 4 Ttt i (14) 5767 4579 1236 tO " (9) 5 6 5 5 u (15) 7647 1239 3789 m ADOITJON, uld be, their ilieir position but also by the addends (16) 6673 123? 2345 (17) 8767 4567 1^34 (19) 5147 3745 6789 (20) 34567 47891 41234 39 (21) 73456 4567? 9123-J 3, by adding iiently plain itil we come ion [Sec. I. al points of ame vertical unchanged ; •004785 •003257 •006546 •0X4588 (22) (23) (24) (25) (26) (27) 76789 34567 78789 34676 73412 36707 46767 89123 01007 78767 - 70760 46770 12476 45678 34667 45679 47076 36767 (28) (29) (30) (31) (32) (88) 45697 76767 23456 46678 23745 87967 87676 45677 78912 91234 67891 32785 36767 76988 34567 66789 23456 64127 (34) (35) (36) (37) (38) (39) 30071 45667 45676 37412 37645 67456 47656 12345 76767 12345 45676 34567 12345 37373 123-15 67891 37676 12345 47676 45674 67891 10707 71267 67891 lution. 8) ^4 4 4 4 «5 " (9) 5 5 5 5 - u (15) 7647 1239 3789 (40) 71234 12498 91379 92456 (46) 87376 12677 88991 23478 (41) 19123 67345 67777 88899 (47) 78967 12345 73707 12(371 (42) 93456 13767 37124 12156 (48) 34567 12345 7776G 67345 (48) 45678 34567 12345 99999 (19) 47676 12345 67671 10070 (44) 45679 34567 12345 76767 (50) 67678 12345 67912 4G7()7 (45) 76766 34567 12345 67891 (51) 67667 34567 23456 76799 40 ADDITION. (52) 76769 12346 76776 466G6 (53) 57667 19807 34076 13707 (64) 767346 4767 of 467007 123456 (55) 478894 767367 412346 671234 (56) 876767 123764 845678 912346 (67) 676 4689 87 84028 (58) 74564 7674 376 6 (69) 5676 1667 63 6767 (60) 76746 71207 100 . 66 (61) 67674 76670 36 77 (62) 42-37 66-84 27-93 62-41 (63) 0-87 6-273 8-127 ?5-63 (64) 03-786 20-766 00-253 10-004 (66) 86-772 6034-82 57-8563 712-62 (66) -00007 -06236 •0572 •21 (67) 6471-8 663-47 21-60:^ 0-00007 (68) 81-0235 376-03 4712-5 6-53712 (69) 0-0007 5000- 427- 37-12 (70) 8453-6 -37 8456-302 •007 (71) 576-34 4000-005 213-5 2763- m: 72. £7654 + £50121 + £100 + £76767 4- £67^5 =£135317. 73. £10 + £7676 + £97674 + £676 + £9017 =£115053. 74. ^971 +£400+£97476+£30+£7000+£76734 =£18261 1. T5. 10000 + 76567 + 10 + 76734 + 6763 + 67674-1 =176842. -r"/D/-M 76. 1 + 2 + 7676 + 100 + 9 + 7767 + 67=15622 /7. 76 + 9970 -f 33 + 9977+100 f 67647 + 676760 =764563. -rJiy>iuu ADDITION. 41 (67) 676 4589 87 84028 (68) 0-87 6 -273 8-127 ?5-63 (67) ri-8 33-47 21-602 0-00007 (71) 576-34 000-006 213-5 753 • + £675 - £9017 ■£76734 6767+1 =15622. -676760 m 78. -75 + -6 + -756 + '7254 +'345 +'5 +'005 +-07 • =3-7514. 79. •4+74-47+37-007+75-05+747-077=:934-004. 80. 5G-054-4-75 + -007+36-14+4-672=101-619. 81. •76 + -0076 + 76 + -5 + 5 + -05.=82'3176. 82. •5 + -05-i--005+5 + 50 + 500=:555'555. 83. •367+56-7+762 + 97-6+471==1387-667. 84. 1+-1 + 10 + '01 + 160+-001=17M11. 85. 3-76 + 44-3+476-l-t-5-5=529-66. 86. 36-77+4'42+M001 + -6=42-8901. 87. A merchant owes to A. £1500 ; to B. £408 ; to 0. £1310 ; to D. £50 ; and to E. £1900 ; what is the sum of all his debts ? A7i,s. £5168. 88. A merchant has received the following sums : — . £200, £315, £317, £10, £172, £513 and £9 ; what ia the amount of all ? Ans. £1536. 89. A merchant bought 7 casks of merchandize. No. 1 weighed 310 tb ; No. 2, 420 ft ; No. 3, 338 ft ; No. 4, 335 ft ; No. 5, 400 ft ; No. 6, 412 ft ; and No. 7 429 ft : what is the weight of the entire ? Ans. 2644 lb. 90. What IS the total weight of 9 casks of goods : Nos. 1, 2, and 3, weighed each 350 ft ; Nos. 4 and 5, each 331 ft ; No. 6, 310 ft ; Nos. 7, 8, and 9, each 342 ft .? Am. 3048 ft. 91. A merchant paid the following sums : — ^£5000, £2040, £1320, £1100, and £9070; how much was the amount of all the payments ? Ans. £18530. 92. A linen draper sold 10 pieces of cloth, the first contained 34 yards ; the second, third, fourth, and fifth, each 36 yards ; the sixth, seventh, and eighth, each 33 yards ; and the ninth and tenth each 35 yards ; how many yards were there in all .? Ans. 347. 93. A cashier received six bags of money, the first hold £1034 ; the second, £1025 ; the third, £2008 ; tho fourth, £7013 ; the fifth, £5075 ; and the sixth, £89 ; how much was the whole sum .? Aiis. £16244. 94. A vintner buys 6 pipes of brandy, containin<r as follows :— 126, 118, 125, 121, 127, and 119 galbns ; how many gallons in the whole ? A/as. 736 gals. 95. What is the total weight of 7 casks, No. 1, con- 42 ADDITION. tainiug, om ib ; No. 2, 725 lb j No. 3, 830 ib • No 4 VaBib; No. 6, 6«7 1b, No. B, 609 ib; and No.' 7,' Js como to r "'«'^1"'"<'"° «o^t ^39, what will 20 S auTetlT' "''fy-«?^™i five thousand Lbm; t,™, ^^-^ ™' 'w thousand seven hundred anl tvm>ty.one ; hfty-s,. thousand seven hundred and seven^- inn \,M n ■!,■ ^''"- 206729644. fo„. .•■n- ""™.'"''I'««s and seventy-one thousand ^ four „,dhons and cghty-six thousand ; two mil ionTMd tweive" „£L a ' "'LXX^eno^u^lir, '™j -venty-two thou.,and, „i„e\„nd e'd a"d twen tlr™^ s:^,^t^^l3^£d5:s^eSS^ four hundred and ninety-one thou.,and. ^^J. 3 8700o' 102 Add together one hundred and sixtv-sevon tl,m, dred 'a d'ii' V 'f T^-r™ 'housatdTZ tt fhi:;^; t ntruV, i'^^ntn^fdirrs ■tno ,\,n ,1 ^ ,, -^^^^'. 3665000. iu,j. Add three tcn-tIionsan<ltlis • fo.tv fn,,,. r tenth, ; live hundredths ; six .hou.a.Uths, ti'ltlenltl "i;! ADDI'IION. 43 «andths ; four thousand aud forty K)no ; twcuty-two, one tenth ; one ten-thousandth. ' Ans. 4107*6r)72. 104. Add one thousand ; one ten-thousandth ; five hun- dredths ; fourteen hundred and forty ; two tenths, three ten-thousandths ; five, four tenths, four tliousandths. Ans. 2445-6544. 105. The circulation of promissory notes for the four weeks ending February 3, 1844, was as follows : — Bank of England, about iE21, 228,000 ; private banks of Eng- land and Wales, £4,980,000 ; Joint Stock Banks of lOngland and Wales, ii;3,446,000 ; all the banks of Scot- land, £2,791,000 ; Bank of Ireland, £3,581,000 ; all the other banks of Ireland, £2,429,000 : what was the total circulation ? Ans. £38,455,000. 106. Chronologers have stated that the creation of the World occurred 4004 years before Christ ; the deluge, 2348 ; the call of Abraham, 1921 ; the departure of the Israelites, from Egypt, 1491 ; the foundation of Solomon's temple, 1012 ; the end of the captivity, 536. This being the year 1844, how long is it since each of these events ? Ans. From the creation, 5848 years ; from the deluge, 4192; from the call of Abraham, 3765; from the de- parture of the Israelites, 3335 ; from tlic foundation of the temple, 2856 ; and from the end of the captivity, 2380 107. The deluge, according to this calculation, occur- red ] 656 years after the creation ; the call of Abraham 427 after the deluge ; the departure of the Israelites, 430 after the call of Abraham ; the foundation of the temple, 479 after the departure of the Israelites ; and the end of the captivity, 476 after the foundation of the temple. How many years from the first to the last ? Ans. 3468 years. 108. Adam lived 930 years ; Seth, 912 ; Enos, 905 ; Cainan, 910 ; Mahalaleel, 895 ; Jared, 962 ; Enoch, 365 ; Methuselah, 969 ; Lamech, 777 ; Noah, 950 ; Shem, 600 ; Arphaxad, 438 ; Salah, 433 ; Hebor, 464 ; Peleg, 239 ; Eeu, 239 ; Serug, 230 ; Nahor, 148 ; Terah, 205 ; Abra- ham, 175 ; Isaac, 180 ; Jacob, l47. What is the sum of all their ages ? . Ans. 12073 years 13. The pupil should not be allowed to leave addition, u ADDITIOIf. until ho can with groat rapidity, continually ad.l any of without hositation or furtL mention of the' numbtV J or instance he Bhould not bo allowed to proceeHhus : 8 nr« ia"'' ^f i ^^ ""^ ' ^^-^ 21 ' ^'- J "«r even 9 a, ci b are 16 ; and a are 21 ; &o. Ho shoJld be able, uTu- raately, to add the following— ' 6638 4768 9342 1Q786 in this manner :--2, 8 ... 16 (the sum of the column • of which 1 IS to be carried, and 6 to be set down) " s! 10... 13; 4,11 ... 17; 10,14... 19. •' ' QUESTIONS TO BE ANSWERED BY THE PUPIL. reduce^d'?^[7]"''''^'"^'''"'^"" those of arithmetic be .2. What is addition .? [3J. tion ? ^3^* ""^^ *^' ''''°''' '^ *^' quantities used in addi- t' WkI ^^\*^' -^S"? °^ ^^'^^*^'^°' ^^^ equality ? [41 ndditir;Utit rs"^^^ ^ ^^^ ^'^' - ^^^ ^^-^^ - ^* Su^* ^^ *^*® ^'^^^ ^°^ addition ? [7] 7. What are the reasons for its different parts > [91 8. J^es this rule apply, at whatever side of the deci- fo^dT[llj"' "' '"^ '' *^^ ^"^"^^^^^« '' ^^ -dded a"e 9. How is addition proved ? [10]. ^ 10. What is the reason of this proof.? [10]. 8UUTRACTI0N. 46 SIMPLE SUBTllACTION. 14. Simpk siibtraction is confined to abslract numbers, and apphuate which consist of but one denomination ^subtraction enablcH us to take one number called' tho subtrahend, from another called the minuend. If anv- tlnng s loft, it is called the excels ; in commercial con- cerns, It IS termed the remainder ; and in the mathema- tical sciences, the difertmce. 15. Subtraction is indicated by —, called the minus, or negative sign Thus 5-4=1, read five minus four equal to one, mdicates that if 4 is substracted from 6. unity is left. » Quantities connected by the negative sign cannot be taken, indifferently, m any order ; because, for example, 6-4 is not the same as 4—5. In the former case the positive quantity is the greater, and 1 (which means + L4J) IS left; m the latter, the negative quantity 18 tlie greater and -1, or one to be subtiactld, still remams.^ To illustrate yet further the use and nature ot the signs, let us suppose that we hmx five pounds and owe four;— the five pounds we hate will be repre- sented by5 and our debt by -4 ; taking the 4 f?om the o, we shall have 1 pound ( + 1) remaining. Next let us suppose that we have only four pounds and owe five ; If we take the 5 from the 4-that is, if we pay a. fkr as we can-a debt of one pound, represented by --1, ^111 still remain ;— consequently 5—4=1 ; but nr.^^' V'"'"''^"'? placed over a subtractive quantity, or one having the negative sign prefixed, aiteTs its value, unless we change_all the^igSs but the first •- thus 5-3+2, and 5—3+2, are not the same thing- ^If -^t^^'* ^ but 5-3+2 (3+2 being considered nowas but one quantity) =0 ; for 3+2=5 ;-therefor« j-3+2 IS the same as 5-5, whifA leaves nothing ; or. m herwords, it is equal to 0. If, however, we cli'an.e all the ai'T'"° ^-^^r^^*- ai.~ x?-_i .i i _ . ' ^""^,0 i- -uc sj^,.,.j vAvcpt ixi-c m-Ki, ine vaiuo of the quantity is l^M 4R ••UHTRACTIOPI. uot aUci-o(l by flit) viiKMilmn ;— thus 5-3-^2=4; and f*~-3 — 2, also, Is c(iual to 4. Again, 27-44-7— 3=27. 27- But ■4+7-3=19. 27—4—74-3 (chaiifyinip all tho ilgng of the ) OT ' ori((iuuI quantitloi, but tha flnt) { ■*■*• • The following examplo will show how the vinculum attecta numbers, according us wo mako it include an additive or a subtractiv<3 quantity : 48-f- 7-3-8-f-7-2 =49. 48-1-7 — 3 — 8 JL7_ 2=49 • what ia under the vinculum beinjf ' additive, it is not necessary to , change any signs. 48-f-7— i5-l-8 — 7-1-2=49 • ^' " ""^ necessary to change aU the d^_L7 q sr~"Tio ,n ',«.'«"» «»'^<'"tlie vinculum. ^'^"r'~'J— "— 7-f-2 =49; it is necessary in this case, alto, 48-1-7-3-8-1-7=2=49; u ^^^^J^^j^^,, ease. In the above, we have sometimes put an additive, and sometimes a subtractive quantity, under the vinculum ; in the former case, wo wore obliged to change the signs ot all the terms connected by the vinculum, except the hrst— that IS, to change all the signs under the vin- culum ; m the latter, to preserve the original value of the quantity, it was not necessary to change any sign. To Subtract Numhers. „nrl!" ^""'^TI: ^^^'' ;^'' ^^«^*« «f *^« subtrahend under those of the same denomination in the minuend— units under units, tens under tens, &c. II. Put a line under the subtrahend, to separate it iroin the remainder. III. Subtract each digit of the subtrahend from tho one over it m the minuend, beginning at the right hand IV. If any order of the minuend be smaller than tho quantity to be subtracted from it, increase it by ten : and cither consider the next order of the minuend as lessoned by unity, or the next order of the subtrahend as in- creased by it. V. After subtracting any denomination of the sub- suiirriACTiOiV. 47 h2=:4; and lie flnt) ! =27. he vinculum include an vinculum hein^ ot aeoessary to 1. to change all the inciilum. thia case, also, ns. in this case. (Iditivc, and I vinculum ; ;e the signs except the Icr the vin- al value of any sign. subtrahend minuend — separate it 1 from the right hand 5r than the J ten ; and IS lessoned end as in- f the sub- trahend from the correspoudiug pjut of the nunuond H',;fc (l.)wii wh.-it i,s loft, if liny thing, in the phiee which' b<.'li)ii(;s to thi! Kaiiie donroiiiri.-itioti (»f the " rcnuiindfr." Vr. ]Jat if th(!rc \a no\\nn<^ Idl, put down a cyphoi-- provided any digit of the " rcMuuiudor" will be niore dis- tant from the deciiuul point, and ut tlio same side of it. 18.. K.\A.MrM.: l.~8ubtr;ict 427 from 71)2. n)2 minuend. 427 «ul;tnilu!iul. u(J5 remainder, ditterenco, or ox<'(!,s«<. Wo cannot take 7 units iunxx 2 unitfl; hut "bomnvin-." nfl It IS calind, one ot the !) tons in the )ninti(Mi(l, ,uid consi,],.,-. «ng It as /6'u unit.s. wo add it to the 2 units, and tbnn liavo 1- units; taking / Iroiii 12 unit.s, 5 arc left:- wo put o in the units pace o/ the "n^maindor." Wo may considor tlio .' tens ot tho niinuend (one luiving been taken away, or borrowed) as 8 tons; or, which is the same thin-. 'may suppose the I tons to remain as they wore, b-.t tho''2 tcn"^ ot tho subtrahend to hi.ve beoomo [\; tlicn, 2 tens from « t.-ns or o tens from U ten.s, and tens are lell :-wo i.ut i\ in the tens' place of tiie "remainder.- 4 hundreds, of tho Hubtrnbend, taken from the 7 Junidreds of the ininurud,' ExAMPLK 2.— Take 5G4 from 7G8. 7G8 504 2U4 When G tons are taken from G ten.s, notliing is Irft : w,. therefore put a cypher in the tens' place of the ^-emainder" KxAMi=j.E 3.— Tako 537 from 5U4. 594 537 s: When 5 hundreds are taken from 5 hun.livds notliln.. 48 SUBTRACTION. Of the subtrahend maybe near those of the mmuend from which they are to be taken ; wo are tlien sure that the JrL^ Sfriuff^Evfl-*'' subtrahend and nainueuS ^^ybe douMn. fo1>,n^^^ "^ arrangement, also, we remove any 1„ ^ \ , *^^ denominations to which the diVits of the sub- trahend belong-their value? being rendered more cert^Lbv S;SZ Tf rf''''' '' the'cligits of th'rnS"' '^ Keason OF H.-The separating line, though convenient i- not of sueh importance as in addition [9] ; si^io th^" remain der » can hardly be mistaken for another quantity Keaso^t of Ill.-When the numbera are considerable powe"' ?tt"mii^^^^^^^^ be effected at once, from'ThflSd poweis ot tne mind; we therefore divide the ffiven ouantitipq into parts; and it is clear that the sum of the d'ffe?en?es of th^ sTmTrS"/ ''T' ^f, ^"l^i^ *^ *»- diffe're^r Ween to 500 4nl7n 9^TS =7*^"'"' ^^^-^27 is evidently equal rebble^"^?"^ w7?.+^~J\r "•''? ^^ «h«^» *« the child by be necesstrv tonU. ^'"^ ^* "ilP^^*. ^^^'^ ''^^' ^^cause it maj oe necessaiy to alter some of the d g ts of the minuend so a*^ to make it possible to subtract from them the corresnondin^ ones of the subtrahend; but, unless we beSn at the STt h3 Bide, we cannot know what alteratioi-s may be iVquired ''''^ thin'tlfo'n '" '^-T^' ",^^. ^"g^* '' tlie iinuend be smaller than the corresponding digit of the subtrahend, we can proceed l?on fTl f *•'" ""T-. .*;"''*' ^« ^'-^y i^^rease .hat denmina- tion ot the minuend which is too small, by borrowing <,nrfrom or'tE v^^Ms t^obf'''^' ""'. 'r f t^e lower InlSaL^ p^|^:ii; i^ tr^^. ^i^^^^r^it^ s idrcds. tens. units. 7 4 8 2 12 = 792, the minuend. / = 427, the subtrahend ^ 5 = 305, the diflference. an^'r,^wV' "I® "['^y «dd equal quantities to both minuend woulfw '"'^' ^'^^"^ ^^" ^°^ ^^^-'- the difference; tien we HuiKireds. tens. 7 9 4 2 + 1 6 units. 2 4- 10 == 792 -f 10, the minuend -f 10. 7 = 42 7 -f 10, tlie subtrahend -f 10. == 365 -|- 0, the same difference. ini«!fKf"'''^ ""a T'!''*'?"-? ^« ^0 not use the given minuend and subtrahend, but others which produce the ^ImTvZZl^T, Reason of V.— The remainders obtained bv subtraetin^ successivfilv. fl.o riiffn«„„4. .i„„_. •, ,. "'^ - "/ suDtracting, froai those which correspond in the minuend are the i^ari of ,1 ij," SUBTRAC'IION. > minuend from that the corres- linueud may be WQ remove any gits of the &ub- nore certain, by s minuend. 1 convenient, is ie the " I'cmain- tity. 3 considerable, pom the limited ;iven quantities e differences of jrence between Jvidontly equal ;o the child by because it may minuend, so as s corresponding the right hand equired. nd be smaller we can proceed ihat denoraina- iwing one from denomination, to those of the we alter the , in the exam* uend. trahcnd. rence. both minuend Jnce ; then we id -f- 10. lend -f 10. ifforence. iven minuend no remainder, subtracting, ts Bubtrahend i the jparts of 4Q the total remainder. They are to be set down under the .Ipnn 20 Proof* of SvMradion.—Add to^-.her the re- mainder and subtrahend ; and the mm sliould be equal to the minuend. For, the remainder expresses by Lw £hf' *t 'f ' r""^"^'^«r to the subtrahend should make it equal to the minuend ; thus 8754 minuend. 6839 subtral. nd. ^ 2915 difference. ) Sum of difference and subtrahend, 8754=.minuend. wh?t^^tf^'^ */f /^"^^^^*»1*^^ f'-*^"^ ^^^e minuend, and f hp I ' 7 -"^^1 ^' "'^"'^^ ^" ^^'' subtrahend. For 8G84 minuend. Pun^r. . aroA ^- i rciQK 1 i 1 , irRooF : obrf4 minuend i98o subtrahend. J549 remafnder. 649 remainder. New remainder, 7985=subtrahend 8034 minuend. 7985 subtrahend. T,.„, , , 049 remainder. Difference between remainder and minuend, 7986= subtrahend T^mV' Tho^lT''-*\'^'^'f'' *■". ^^'^^'^■f^^^^^ contain Deci^ s do "i^fl r • ^T^ ^''^'V ^PPli^'-^t'J^, at whatever k fold tv'TnP'"'^ '"^^ '' '"^"^'^^ ^^^« ^i^^^ts may VMT '7.;' follows, as in addition [11], from the Y'y n.Uuie of notation. It is necessary to put th.= decimal point of the remainder under the IZS'JnU v^lZ.lT! fl "; '""S ^^ ^^'""^ ^*"-^^t, have the samo ^alue as the digits from which they have been derived. 60 Example. SUBTRACTION. -Subtract 427-85 from 503-04. 663-04 427-85 135-19 Since the digit to the right of the decimal point in the onthT' ?■' ''"'^''''''' '"^"* '' ^'^'' ''^''' *^»« subtraction o i.o t e inhtl n' T^""^ P?mt indicates vvimt remuins^fter the subtraction of tlie units, it expresses so many units:- all this IS shown by the position of the decimal point. ' An^^'J\ ^y^'''^'' ^'''™ *^^ principles of notation [Sec. I. 40J, that however we may alter the decimal points of the mmuend and subtrahend, as long as the/stand in the same vertical column, the didts of the difference are not changed ; thus, in the following examples, the fiame digits are found in all the remainders •— 4362 8547 815 436 354' 81-6 43-62 35-47 8-15 •4362 •3547 •0815 •000 J 3 62 •0003547 •0000815 EXERCISES IN SUBTRACTION. . From Take (1) 1969 1408 (2) 7432 6711 (3) 9076 4567 (4) 8146 4377 (5) 3176 2907 (fi) 76877 45761 Froi.i Take (7) 86167 61376 (8) 67777 46699 (9) 71234 43412 (10) 900076 899934 (11) 376704 297610 (12) 745674 376789 From Take (13) 67001 35690 (14) 9733376 4124767 (15) 567i)74 476476 (16) 473(i76 S21799 (17) 6310756 3767016 (18) 376576 240940 m SUBTRACTION. il point In the )ti-acti(m of tho ice the digit to i remsiiiis after many units; — al point. :ation [Sec. 1. nal points of they stand io ;he difference 3xamples, tlie 3 : — •0001362 •000;J547 •0000815 (19. From 345070 Take ]799 (20) 234100 9U0 (21) 4367676 25G560 (22) 845073 I2479'J (23) 70101076 37091734 61 (24) 67300000 31237777 From Take (25) (26) (27) (28) r29) no^ S?? JS f= ??S = ~ 47134777 1123640 7476909 (31) From 7045076 Take 3077097 (36) From 11000000 Take 9919919 (82) 87670070 26716645 (37) 3000001 2199077 (33) 70000000 9999999 (38) 8Q00800 (J77776 (34) (35) 70040500 60070007 56767767 41234016 (39) 8000000 62358 (40) 404006b 220202 (5) (6) n-Q 76377 i907 45761 I) 704 610 ) (12) 745674 376789 (18) '50 370570 )16 240940 From Take (41) 85-73 42-16 (46) From 0-00003 Take 0-00048 (42) 805 -4 73-2 (47) 874-32 6-63705 (43) 694-763 85-600 (48) 67-004 2-3 (44) 47-030 0-078 (49) 47632- 0-845003 (45) 52-137 20-005 (50) 400-327 0-0006 745676—507456=178220 500789—75074=501115.* 941000-5007=935993. 9/001—50077=40024 70734-977=75757. 56400-100=50300. 700000—99=099901. 5700—500=5200. 9777—89=9088. 70000-1=75099. 90017-3=90014. 02. 97777-4=97773. 03. 00000-1=59999. 64. 75477—76=76401. 65. 7-97- 1-05=6-92. 00. 1-75— .074=1-676. 07. 97-07—4-709=92-301. 08. /• 05— 4-776=2-274. 09. 10-701—9-001=1-76. 70. 12-10009-7-121=4-07909 n. 170-1 — •007^-..176-093.' 72. 15-00 -< -803=7-197. 53 SUBTRACTION. 73. What number, .iddcd to 9709, will make it 10901 oaJ: ^J^^*"*^^' ^"^1^t 20 pipes of hrandj, containing 2459 gallons, and sold 14 pipes, containing 1680 gal- Ions ; how many pipes and gallons liad he remaining ? Ans. 6 pipes and 779 gallons. 75. A merchant bought 664 hides, weighin^r 16800 Jb, and sold of them 260 hides, weighing 78091b ; how many hides had he unsold, and what was their wei^rht > 76. Am. 304 hides, weighing 8991 lb A gentleman who had 1756 acres of land, gives 2o0 acres to his eldest, and 230 to his second son ; how many acres did he retain in his possession > Ans. 1276 77. A merchant owes to A. i^SOO ; to B. £90 • to D |7o0; toD. ^600. To moot tl.co' debts t has but d;.971 ; how much is he deficient ? Ans jei269 78 Paris is about 225 English miles distant from London; Eonie, 950; Madrid, 860; Vienna, 820- Copenhagen, 610; Geneva, 460 ; Moscoav, 1660 : Gib- ral^nr, 1160; and Constantinople, 1600. How much more distant IS Constantinople than Paris; Rome than Madrid ; and Vienna than Copenhagen. And how much less distmit IS Geneva than Moscow; and Paris than Madrid } Am. Constantinople is 1375 miles moro dis- taut than Pans; Rome, 90 more than Madrid; and V lenna, 210 more than Copenhagen. Geneva is 1200 miles less distant than Moscow; and Paris, 635 less tnan iuadnd. 79. How much was the Jewish greater than the J^-.ngIish mile ; allowing the former to have been 1-3817 miles Endish > 80. mile ; mile 5 81. Am. 0-3817. How much IS the English gi-eater than the Roman allowing the latter to have been 0-915719 of a T#u'^-' .1 , . ^^^- 0-084281, W hai IS the value of 6 - 3 + 1 5 - 4 .? Am. 1 4 Afis S'? Of 47-6-2+1-244-16--34 } Am. 52 94 84. What is the differencc betwccn 15+13—6—81 + Am. 38. S2. Of 43 + 7-3^^? 83. 84. __ ^ 02, and 15+13—6=11 + 62 } 23. Before leaving this rule, the pupU should "be able ;j i M "* MULTIPilCATlON. 53 :c it 10901 Alls. 1192, 5 containing ; 1680 gal- :nainiug ? '79 gallons, iing 16800 09 It) ; how eir weight ? ig 8991 lb. land, gives I son ; how Ans. 1276. £90 ; to C. be has but ns. jei269. stant from ma, 820 ; 660 ; Gib- low much lome than how much Paris than moro dis- Jrid ; and a is 1200 , 635 less than the an 1-3817 . 0-3817. iie Roman )719 of a )-084281 Ans. 14 Ans. 33 IS. 52 94 •6—81 + Ans, 38. I be able to take any of the nine digits continually from a given number, without stopping or hesitating. Thus, sub- tracting 7 from 94, he should say, 94, 87, 80, &c. ; and should proceed, for instance, with the following exampla 5376 4298 1078 m this manner :~8, 16.. .8 (the difference, to be set down); 10, 17...7; 3,3...0; 4, 5...1. QUESTIONS TO BE ANSWERED EV THE PUPIL. 1. What is subtraction .? [14]. 2. What are the names of 1*e terms used in subtrac- tion ? [14]. 3. What IS the sign of subtraction ? [15]. 4. IIow is the vinculum used, with a subtractive quantity? [16]. 5. What is the rule for subtraction .? [17]. 6. What are the reasons of its different parts.? [19]. 7. Docs it apply, when there are decimals ? [211 tion 8. How is subtraction proved, and wliy .? [2oT. 9. Exemplify a brief mode of performiu'^ su " ' [23] ^ ° subtrac- SIMPLE MULTirLIGATION. 24. Simple multiplication is confined to abstract numbers, and apphcate which coataiu but one denomi- nation. ,».^?r ^^'?^'"" '"f '^'' "' f" ''^^^ ^ ^"^'^ity. ^'^^"cd the ^ ¥^anrl, a number of times indicated by the ,nM- ittl U ^''f'^'^''' .'% t'^^t h wlucli we multiply : the x^sult m he multiplication is called the jJdt. addo d,'' ,n multipheatiou, is termed the <' multipli ' JZh^ !^' '^^ '' designatoa tho " product.'' The Uuautu.es which, whou muitipljcd log.^l|cr, give tho ' ' J.) m Ml 64 MULTIPLICATION. poduot, nro cnlle.l also factors, nnd, when they ,re •S'^rtf^^'"* J^^^'^^ ;"/^-^ ^- — than' two actors m that case, the multiplicand, niultinlier or b and 7, be the factors, either 6 times 6 may be con ^^V^^;/""Itiphcand, and 6 times 7 as the mulLLr 2o. Quantities not formed by the continued addS^on of any number, but unity-that is, which are not h^ product^; of any two numbers, unle'ss unityTs taken as one of them-are called privii numbers : Yll ot ts are termed co^nposiie. Thus 3 and 5 are p me but 9 and 14 are composite numbers; because oniyXJ midtipbed },y...,,will p«pduce "'three," aU oKi mu tip led by one, will produce " five, "-but //S multi^hed by three will produce " nine," ind seTeSi mil tiphed by hvo will produce " fourteen " fl!.r. \ ■*PT^'^ ^^ ^" e«^%-er-or, in other words matder i^ f'^^r^f ^^"^^ ^^ ^^^^^^^ leaving a rei contabed in I .1 "" '\- ^""'^^'^ "^ ^^^ ^'^^^'^ it i« Pd from if a^t^b^^f r^^pi-eS b'yNt" measure ot 14, because, taking t as often as r^n^^ihC. from 14 4 will .till bo lefti-this, I.3_3=,0, /o-S -5=0, but 14—5=9, and 9—5=4. Measure, ', ,:. , ' "," ^^ — '-'^ry, una y — 5=4. submultiple, and aliquot part, are synonymous. is V numllrTrr V'''' '^ *^" ^^ ^^°r« q^^^tities IS a number that will measure each of them • it is a measure comnon to them. Numbers which^ have no ocner all otliers are comj^^^z^e to each other. Thus 7 and 5 are i;r^/;^e to each other, for unity alone will b::s:3':iiV ' ^^^^ ^^^- ^^^^^sr^ :^,^^ uecause 3 wUl measure either. It is evident that two ITi ^Z^'l ""f "= p™- "> -^''' *" , ttfs e°ceM „n^t. fr"' •""''''"'■'', •'"™"' "»• '^ ""•'-■. ''"<'- MUI,TIPLICATIO^^ 65 on they are I'e than two lultiplier, or Thus, if 5 May be con- iiltiplier — oi nultiplier. led addition ire not the is taken as I others are ime, but 9 only tkreey id onlyfivey -but, three I seven mul- tne number thor words, ving a ro- ot pari of 3cause it is Q be sub- d by 3, an ) is not a IS possible I, JO— 5r= Measure, • quantities 1 : it is a have no 'Me to each Thus 7 tlone will ch other, that two r thus 3 :o, and — will niea- ■"fti Two numbers may bo oninpnsito to each other, and yet (yne of them may be a fvme number ; thus 5 and 25 are both measured by 5, still the former is ytrim. Two numbers may be composite, and yet prime to emh other ; thus 9 and 14 are both composite numbers, yet they have no covwion measure but unity. 28. The greatest common measure of two or more numbers, is the greatest number which is their common measure ; thus 30 and 60 are measured by 5, 10 15 and 30 ; therefore each of these is their ccmimon mea- sure ; — but 30 is their greatest common measure. When a product is formed by factors which are integers, it is measured by each of them. 29.^ One number i- the i^uUiple of another, if it contam the latter a number of times expressed by an integer. Thus 27 is a multiple of 9, because it con- tams It a number of times expressed by 3, an integer Any quantity is the multiple of its measure, and the measure of its multiple. ^ 30. The com7non multiple of two or more quantities, IS a number that is the multiple of each, by an intcffer •-- thus 40 IS the common multiple of 8 and 5 ; since it is a multiple of 8 by 5, an integer, and of 5 by 8, an integer. ^ LhY<^st common multiple of two or more quantities, IS the /m5^ number which is their common multiple--, thus 30 IS a common multiple of 3 and 5 ; but 15 is then- least copunon multiple ; for no number smaller tnan lo contains each of them exactly. 31. The equivmltiples of two or more numbers, are then- products, when multiplied by the same number ;— tlius 27, 12 and IS, are equimultiples of 9, 4, and 6 • because, multiplying 9 by three, gives 27, multiplying 4 ^^4f 'tV^u-'r^-'.''^^ ^^^itiplying 6 by three, give^ IS. S2 Multiplication greatly abbreviates the process of addition ;— for example, to add 68965 to itself 7000 times })y audition," would be a work of great labour, and con- sume much time ; but by " multiplication," as we shall find presently It cn^i be done with case, in less than a minute. roi.i 7 i?v "•''^ .'"'"' inaccurate, to have stated L-^J that multiplication is a species of addition ; since we can know the product of t^vo quantities without havin^' 06 MULTIPLICATION. recourse to that rule, if tliey are found in the multipli- cation table ?..t it must not be forgotten that the mul- plioation table IS actually the result of additions, long since made ; without its assistance, to multiply so simplf a number as 4 by so smaU a one as five, we shodd be obliged to proceed as follows, 4 4 4 4 4 20 performing the addition, as with any other addends The multiplicat on tabl^is due to Pythagoras, a" cele- bSortcS '''''^''''^ ^^^ ^- ^-^ ^^^ y-s 34. We express multiplication by X ; thus 5x7— thatXT *^\^r^*f-d by 7 ire equal tol's, ^^ that the product of 5 aTid 7, or of 5 by 7, is equal to 35 When a quantity under the vinculum 'is toTe muf^ plied-for, to multiply the whole, we must multiply eack of Its parts^:— tlms^^7+8=3=3X7+3xS-l 3X3; and 4+5X8+3-6, means that each of the terns under thelaiier vinculum, is to be multiplied by each of those under the former ^ ^ maf be^ro^ltf ""'''''"' f ^^ ?^' ^^^n of multiplication may be lead in any order; thus 6X6=6X5 This Will be evident from the foUowing mustration, by which of it :-! ' ^ ' ^'^^^^^"S to the view we take «> 8 ♦ ♦ • ♦ ♦ • ♦ • • » • » "^ ^ » • Quantities connected by the mgn of multipUcation, MULTIPLICATION. Jie multipli- bat the mul- tlitiona, long ly so simple e should be 67 Idends. >ras, a cele- 590 years us 5x7= I to 35, or qual to 35. ) be multi- be multi- t multiply r+3xs— ch of the Jtiplied by ttiplication <5. This . by which Bonsidered y we take V 't plicati^ ion. pre multiplied if we multiply one of the factors ; thus GX7X3 multiplied by 4=6X7 multiplied by 3X4. 36. To prepare him for multiplication, the pupil should be made, on seeing any two digits, to name their product, without mentionhig the digits tiiemselves. Thus, a largo number having been set down, he may begin mth the product of the first and second digits; and then proceed with that of the second and thh-d, &c! Taking 587C349258G7 for an example, he should say: — 40 (the product of 5 and 8) ; 56 (the product of 8 and 7) ; 42 ; 18 ; &c.,as rapidly as he could read 5, 8, 7, &c. To Mibltijply Nmibers. 37. When neither multiplicand, nor multiplier ex- ceeds 12 — Rule. — Find the product of the given numbers by the multiplication table, page 1. The pupil should be perfectly familiar with this table. ^ Example.— What is the product of 5 and 7 ? The mul- tiplication table shows that 5x7=35, (5 times 7 are 35). 38. This rule is applicable, whatever may be the relative values of the multiplicand and multiplier ; that is [Sec. I. 18 and 40], whatever may be the kind of units expressed— provided their ahsolwte values do not exceed 12. Thus, for instance, 1200X90, would come under it, as well as 12X9 ; also •0009X0-8, as well as 9X8. We shall reserve what is to be said of the man- agement of cyphers, and decimals for the next rule ; it will be equally true, however, in all cases of multiplica- tion. 39. When tlie multiplicand does, but the multiplier does not exceed 12 — Rule. — I. Place the multiplier under that denomi- nation of the multiplicand to which it belongs. II. Put a line under the multiplier, to seplirate it from the product. ni. Multiply each denomination of the multiplicand by tiie multiplier— bogiuniii;;^ ut the rit^'ht hand side. --+1 08 MULTIPLICATIOX. I\. If tlm prodiipt of tho multiplier and any digit of tho liiultiphcand is Ichs than ten, set it down under that (tigit ; but if it bo greater, for every ten it contains carry one to tho next produ.., and ])ut down only what remains, after d. o' u tir. the tens; if nothing remains, put down a cypher. ' ,V. Set down the last product in full. 40. Example. 1.— What ia the product of 897351x4? SOTI^Sl multiplicand. 4 multiplier. 3581)404 product. 4 times one unit are 4 units; since 4 is less than ten, it gives nothing to be "carried," we, therefore, Bet it down n the units' place r f the product. 4 times 5 are twenty (tens)-: which are equal to 2 tens of tens, or hundreds to I o carried, and no units of tons to be set down in the tens' place of the product— in which, therefore, we put a cypher 4 times 3 are 12 (hundreds), which, with the 2 hundids to bo carried from the tens, make 14 hundreds; these are equal to one thousand to bo carried, and 4 to be set down in the thousan<l8' pluee of the product. 4 times 7 are 28 (thou- sands), and 1 thousand to be carried, are 29 thousands ; or 2 to be carried to tho next product, and 9 to be sot do\vn 4 times 9 are 3b, and 2 are 38 ; or 3 to be carrriod, and 8 to be set down 4 times 8 are 32, and 3 to be carried are 35 ; which 13 to be set down, since there is nothing in the next denomination of the multiplicand. Example 2.— Multiply 80073 by 2. 80073 2 16014G Twice 3 units are units ; G being less than ten, gives nothing to be carried, hence we put it down in the units' place of the quotient. Twice 7 tens are 14 tens; or 1 ' undrod to be carried, and 4 tens to be set down. As there are no hundreds in the iiadtiplicand, we can have none in the pro- duct, except that whicli is derivtsl from the multiplication ot the tens ; we accordingly put the 1, to be carried, in the hundreds' place of the product. Since there are no thou- sands in the multiplicaud. nor any to be carried, we put a cypher in that denomination of tho product, to keep any significant iigures that follow, in their proper places. i i MULTIPLICATION. 5U 41. Reason of I.— Tho multiplier ia to ))o pluccd under that dcnominfiMon of the multiplicaud to wliicli it belouRs; sinco- tliere is tJien no doubt of its vhIuo. Sometimes it is necessary (0 add cypiiers in putting down the muiti|il:er ; thu.s. EXAMPI.E 1.— 478 multiplied by 2 liundred— 47H multiplicand. 200 multiplier. Example 2.-539 multiplied by 3 ten- thousandths— 68'J • multiplicand. 0-0003 multiplier. Reason of II.— It ia similar to that given for tlio separatinff line in subtraction [10]. ^ e Reason ov III.— Wlien tho multiplicand exceeds a certain amount, the powers of the mind are too limited to allow us to multiply it at once ; we therefore multiply its parts, in suc- CQSsiun, un.l add the results as wo proceed. It is clear that tho sum of the products of the parts by the muliinlior, is equal to the product of tho sum of tlie parts by the same multi- plier :— tlius, 537x8 is evidently equal to 500 x8-f;;0x 84-7x8 For multiplying all the parts, is multiplying the wliole ; since the whole is equal to the sum of all its parts. We begin at the rigl.^ hand side to avoid the necessity of athnimrds adding together the subordinate products Thus taking the example given above ; were wo to begin at the left liand, the process would be — 897351 4 3200000=800000x4 360000= 90000X4 28000^ 7000x4 1200= 300x4 200= 50X4 4= 1x4 3589404=8um of products. iV^Z^ri °^ jy:~^'r'^,*^'® ''""^ ^'' ^^''"^^ "^f "'« fourth part of the rule for addition [9]; the product of the multipl/er and any denomination of the multiplicand, being equivalent to the bum of a colur. n m addition. It is easy to change the o-iveu ixainp e to an .xercise in addition; for 807851 x I, is theime thing as 897351 897331 897351 897351 3589404 m J fl 60 hlULTirUCATlOS. RKABopr OF y.-It follows, that tho Inflt pro^liict h to be eot <lown in ful; tor tlie tens it contains will not bo incroaseU : they in«y, tlioroloro, bo sot down at once. This riilo includca all casos in wlilnli tho ahsolii/e value ^ of the di^'its in the luultiplior d.x-s not excoea , 12. Their rcdativo value is not niatori-ii ; for it is as easy to multiply by 2 thousands as by 2 units. 42. To prove multiplication, wluni tho mnltiplier dooa not exceed 12. Multiply the multiplionnd by th(> mul- tiplior, minus one ; and add the multlplicjind to the pro- duct. Tho sum should bo the same as the product of tho multiplicand and multiplier. Example.— Multiply G432 by 7, and prove tho i-viult. C432 multiplicand. 6=7 (the multiplier) ~1 6432 3S502 multiplicand xO. 7(=C+1) 0432 multiplicand Xl. 45024 = ' 45024multipllcandmultip]iodby 0,1=7. We have multiplied by 0, and by 1, and adtlod the results ; but SIX times the multiplicand, plus once the multiplicand, IS equal to seven times tho multiplicand. What we obtain from the two processes snould be the same, for we Wve merely used two methods of doing one thino-. EXERCISES FOR THE PUPIL. Multiply Bj (1) 76762 2 (5) 763452 6 (9) 866342 11 (2) 67450 2 (6) 456769 7 (3) 78976 6 (7) 854709 8 (4) 57340 6 Multiply By (8N 45678f ? Multiply By (10) 788679 12 (H) 476387^ 11 (12) fa>t29763 12 MULTIPLICATION. 61 5t ?fl to be Bot )o incronsed : ;ho ahsolnfe not (jxcood for it is as Itiplier dooa ly the HI al- io tlio pro- product of I'Viult. th» reanlts ; ultiplican<l, i we obtain 3r we Wve (4) 57040 6 43. To Multiply when the Quantities contain Cyphers, or Dmrnals. — Slie rules alroady given aro applicable ; those which follow aro consetjuonces of them. When thoro arc cyplicrs at the cud of tho multipli- cand (cyphers in tho middle of it, Lavo been already noticed [40])— Rule. — Multiply as if there were none, and add to tho product as many cyphers as have boon neglected. For Tho greater tho quantity multiplied, tho grontor ought to be tho product. Example. -Multiply 5G000 by 4. 5C00O 4 224000 4 timoa imita in tho fourth place from the decimal point, arc evidently 24 ixnits in the same place ; — that is, 2 in tha fiflh place, to be carried, and 4 in the fourth^ to be set down. That wo may leave no doubt of the 4 being in tho fourth }>lace of" tho ;or()duct, we put three cyphers to tho rij^ht land. 4 times G are 20, and tho 2 to be carried, make 22. 44. If tho multiplier contains cyphers — Rule. — Multiply as if there were none^ and add to the product as many cyphers as have been neglected. Tho greater the multiplier, tho greater the number of times the multiplicand is added to itself; and, therefore, the greater the product. ExAMPLK.— Multiply 507 by 200. 5G7 200 113400 From what we have said [35], it follows that 200x7 is the same as 7x200 ; but 7 times 2 hundred are 14 hundred ; and, consequently, 200 times 7 are 14 hundred ;~that is, 1 in tho fourth place, to be carried, and 4 in the third, to be set down. We add two cyphers, to show that the 4 is in the third place. 45. If both multiplicand and multiplier contain cyphers — Rule. — Multiply as if there were none in either, and add to the product as many cyphers as are found in both. d2 m 62 MULTirLICATiON. Each of the quantities to be multiplied adcla cyphers to tho product [43 and 44]. Example.- Miihiply 46000 hy 800. 40000 800 50800000 _ 8 times G thousand yrocld bo 48 times six thousand ought to prod thousand number 8 hundred 100 times greater— or 48 hundred thousand ;— that is, 4 in the scvcnt/i place from the decimal point, to be carried, and 8 in tlie are required. But, 5 cyph xLxtk place, to be set down. to keep the 8 in the sixth place. After ascertaining the position of the first digit in the p.^duct— from what the pupil already knows— there cjin be no difficulty Avith tho other digits. 46. When there are dechnal places in the multipli- cand — Rule. — ^i\Iultiply as if there were none, and remove the product (by nieails of the decininl point) so many places to the right as there have been docuuals neglected. Tlie smaller the quantity multiplied, the loss the product Example.— Multiply 5-07 by 4. 5-67 4 22-08 4 times 7 hundredths are 28 hundreths :— or 2 tcntlis, to bo carried, and 8 hundredth « — or 8 in the second place, to the right of tho decimal point, to be set down. 4 times 6 tenths are 24 tenths, which, with the 2 tenths to be carried, make 20 tenths ; — or 2 units to be carried, and G tenths to bo set down. To show that tlie re[>rosents tenths, we put the decimal point to tho left of it. 4 times 5 units are 20 wiits, wliicli, with the 2 to ])e carried, make 22 units. 47. When there are decimal:? in the multiplier — Rule. — Multiply as if there wore none, and remove the product so many places to the right as there are decimals in the multiplier. The smaller tho quantity by which we multiply, the less must be the rwult. )hers to tlio : 8 hundred 100 times the seucnt/i d 8 in tlie e required, iiining the 1 what tlie 7 Avith tho 3 muUipli- •cmove the any places cted. product I tcntlis, to d place, to 4 times 6 he carried, 3 tenths to the, we put nJtH are 20 nits. ier — id remove there are y, the less Example.- MULTIPLICATION, -Muiaply 5Go by -07 503 007 63 39-41 3 multiplied hj 7 hundredths, is the same [351 as 7 hun- dredths multiplied by 3 ; whioh is equal to 21 hundredths : — or 2 tenths to be carried, and 1 hundredth — or 1 in the second place to the right of the decimal point, to be set down. Of course the 4, derived from the next product, must be 07ie place from the decimal point, «;c, 48. When there are decimals in both multiplicand and multiplier — Rule. — Multiply as if there were none, and move the product so many places to the right as there are decimals in both. In this case the product is diminished, by the emallnesB of both multiplicand and multiplier. Example 1.— Multiply 56-3 by -08. * 56-3 •08 4-504 8 times 3 tenths are 2*4 [46] ; consequently a quantity one hundred times less than o — or -08, multiplied by three- tenths, vrill give a quantity one hvmdred times less than 2-4— or -024 ; that is, 4 in the third place from the decimal point, to be set dowTi, and 2 in the second place, to be carried. Example 2.— Multiply 5-63 by 0- 00005. 5-63 0-00005 0-0002815 49. When there are decimals in the multiplicand, and cyphers in the multiplier 5 or the contrary — Rule. — Multiply as if there were neither cyphers nor decimals ; then, if the decimals exceed the cyphers, move the product so many places to the right as will be equal to the excess ; but if the cyphers exceed the deci- mals, move it so many places to the kft as will be equal to the excess. ' The cyphers move the product to the left, the decimals to the right ; the effect of both together, therefore, will be equal to the difference of their separate effects. 64 MULTIPLICATION. PI ExAxMPLE 1.— Multiply 4600 bv "06 4000 ^ 006 2 cyphers and 2 decimals J excess -=0 276 Example 2.— Multiply 47-63 by 300. 47-63 "^ 300 2 decimals and 2 cyphers; excess =0. 14289 Example 3.— Multiply 85-2 by 7000. _J^^ 1 decimal and 3 cyphers ; exce8fl=2 oji>Men 596400 Example 4.— Multiply 578-36 by 20. 578-35 ^^ _ 2 decimals and 1 cypher; excess =1 decimal. 11567-2 Multiply By EXERCISES FOR THE PUPIL (13) (14) 48960 76460 5 9 (15) 678000 8 (16) 57d00 6 Multiply By (17) 7463 80 (18) 770967 900 (19) 147005 4000 (20) 661*76748 SOOOO Multiply (21) 743560 800 (22) 534900 SOOOO (23) 60000 300 (24) 86000 6000 Slultiply By (25) 62736 o (26) 8 -7563 4 (27) •21875 (28) 0-0007 8 MULTIPLICATION. 05 ss — Multiply By (29) 5G341 0-0003 (30) 85G37 0-005 (31) 721*58 0-0007 (32) 217G-38 0-06 3=0. 2oyj!i.ewi L decimal. (16) 67000 6 (30) o6{t76748 30000 (24) 86000 5000 (28) 0-0007 8 Multiply By (83) 875-432 0-04 (34) 78000 0-3 (35) 51-721 GOOO- (36) 3*^ 0-00007 •00224 In the last example we are obliged to add cyphers to the product, to make up the required number of decimal places. 50. When both multiplicand and multiplier exceed 12— KuLE.— I. riace the digits of the multiplier under those denominations of the multiplicand to which they belong. II. Put a line under the multiplier, to separate 'u from the j)roduct. III. Multiply the multiplicand, and eack part of the multiplier (by the preceding rule [39]), beginning With the digit at the right hand, and taking care to move the product of the multiplicand and each sncce.ssive digit of the multiplier, so mnny places more to the left, than the preceding pi-oduct, as the digit of the multiplier winch produces it is more to the loft tlian the signifi- cant figure by which we have kusi multiplied. IV. Add together all tlie products; and their sum will be tlie product of the multiplicand and multiplier. 51. ExAMPLi:.— Multiply 5634 by 8073. 5034 8)73 lG002=prodact by ?,. 39438 =pro(lact l)y 70. 45072 =product by 8000. 45483282=product Ijy 8073. The product of the nuiUiplicand by 3, requires no e^i^ nation. ^ 66 MULTIPLICATION. 7 tens times 4, or [35] 4 times 7 tens arc 28 tens : — 2 hun- dreds, to be carried, and 8 tens (8 in the second place from the decimal point) to be set down, &c. 8000 times 4, or 4 times 8000, are 32 thousand : — or 3 tens of thousands to be carried, and 2 thousands (2 in the fourth place) to be set down, &c. It is unnecessary to add cyphers, to show the values of the first digits of the different products ; as they are sufficiently indicated by the digits above. The products by 3, by 70, and by 8000, are added together in the ordic-y way. 52. Reasons of I. and II. — They are the same as those given for corresponding parts of tlie preceding rule [41]. IIEASON OF III. — We are obliged to multiply successwely by the parts of the multiplier ; since wo cannot multiply by the whole at once. xlEAsoisr OF IV. — The sum of the products of the multipli- cand by the parts of the multiplier, is evidently equal to the ■product of the multiplicand by the wliole multiplier ; for, in the example just given, 5634 X 8073 = 5684 X 8000 -f- 70 -f 8= [34] 5034 X 8000+5634x70-1-5634x3. Besides [35], we may consider the multiplicand as multiplier, and the multiplier as nmltiplicand ; then, observing the rule would be the same thing as multiplying the new multiplier into the diiFerent parts of the new multiplicand 5 which, we have already seen [41], is the same as multiplying tlie whole multiplicand by the multiplier. The example, just given, would become 8073X5634. 8073 new multiplicand 5684 new multiplier. We are to multiply 3, the first digit of the multiplicand, by 6634, the multiplier; then to multiply 7 (tens), the second digit of the multiplicand, by the multiplier ; &c. When the multiplier was small, we could add the different productti as we proceeded; but we now require a separate addition, — whicii, however, does not affci the nature, nor the reasons of the process. 53. To p'ove multiplication, when the multipliei ex- ceeds 12 — EuLE. — Multiply the multiplier by the multiplicand ; and the product ought to be the same as that of the multiplicand by the multiplier [35] . It is evident, that we could not avail ourselves of this mode of proof, in tho last rule (b[42j ; as it would have supposed the pupil to be then able t-o multiply by a quantity greater than 12 th de lei in of Til ini 7- ha or '■/ aol « th( '4 Tal 1 bc^ 1 'I 3 : — 2 hun- 3laco from lies 4, or 4 iauda to be ) to be set show the i] as they e products e ordic-y e as those [41]. uccessively aultiply by e multipli- jual to the er ; for, in f 70+8= )], we may ultiplier as the same e diiFerent ready seen plicand by Id become plicand, by the second When the )roductt as n, — whici., ions of the tipliei ex- tiplicand ; lat of the dent, that oof, in tho B pupil to • than 12 MULTIPLICATIOINr. 67 mR 54. We may prove multiplication by what is called " casting out the nines." Rule.— Cast the nines from the sum of the digits of the multiplicand and multiplier ; multiply ti-e remain- ders, and cast the nines from the product :— what is now left should be the same as what is obtained, by cast- ing the nines, out of the sum of the digits of the product of the multiplicand by the multiplier. I^.XAMPLE 1. — Let the quantities multiplied be 942G and 'I'aking the nines from 9426, we get 3 us remainder. And from 3785, we get 5. 47130 75408 3x5=15, from which 9 C5982 beino; taken, 28278 • 6 are left. Tiiking the nines from 35077410, 6 are left. The remainders l)e!ng equal, we are to presume tlie multiplication is correct. Tlxe same result, however, would liave been obtained, even if we had misplaced digits, added or omitted cyphers, or fallei. into errors which had counter- acted each other : — with ordinary care, however, none of these is likely to occur. ExAMPr.K 2.— Let the numbers be 70542 and 8436. T:>„king the :aincs from 76542, the remainder is G. Taking them from 8436, it is 3. 459252 229626" 6x3=18, the 306108 remainder from which is 0. 612336 Taking the nines from 645708312 also, the remainder is 0. Tho remainders being the same, the multiplication may be considered right. Example 3.— Lot the numbers be 403 and 54. ^ From 463, the remainder is 4. From 54, ',-■ ' - 1852 4x''-=0 from which the remainder is 0, 23 15 From 2bd02 the remainder is 0. u kl % 68 MULTIPLICATION. Tlie remainder being in each case 0, wo arc to suppose that the multiplication is correctly performed. This proof applies whatever be the position of the decimal point in either of the given numbers. 55. To understand this rule, it must be known that a number, from which 9 is taken as often as possible, will leave the same remainder as will be obtained if 9 be taken as often as possible from the Bum of its di.^its " Since the pupil is not supposed, as yet, to have learned divinon, he cannot use that rule for the purpose of casting out the nines ;- nevertheless, he can easily ellect this object. •^ K !f, o'" f^7^'\ ^^""^^'er be 5C3. The sum of its digits is +,.+'^' ^hile the nvimber itself is 500-fG0+3. First, to take 9 as often as possible from the sum of //,« (hgrls. 5 and 6 aro'll ; from which, 9 being taken, 2 are loit. ^ and 6 are 5, which, not containing 9, is to be set down as the nmainder. Next, to ta^o 9 as often as possible from the mmbcr itself.. 503^=500 + 00+3=5 xl00+Gxl0+3=5x9iq^+Gx 9+1+3,= (if we remove the vinculum [34]), 5x99+5+ Ox.i+b+3 But any nnmber of niacs, will be found to he f tie product of the same number of ones by 9 .-—thus 999— 111X9; 99=11x9; and 9=lx9._Hence 5x99 express;^ a certain number of nines-being 5x11x9 ; it may, there- lure, be cast out; and for a similar reason, Gx9: after wliich there will then be left 5+G+3-from w'hic^i the luues are still to be rejected; but, as this is the sum of the dibits we must, in casting the nines out of it, obtain the same remain- aer as before. Consequently "we get the same remainder whether we cast the mnes out of the number itself, or out of the sum of its digits." Neither the above, nor the following reasoning can offer any difficulty to the pupil who has made himself as fainiiar with the use of the signs as he ought :- they will both, on the contrary, serve to show how much simpbcity, is derived from the u«e of characters express- ing, not only quantities, but processes ; for, by nieanj ot such characters, a long series of argumentation mav be seen, as it were, at a single glance. 5G "Costing the nines from the factors, n-.iripiyina- tU resulting remainders, and casting tlie nines from thiH product, MULTIPLICATION. 09 to suppose ion of the nown that 8 possible, aincd if 9 its digits." ve learned urpose of 3aa easily ;s digits is sum of it.<i ken, 2 are to he set mhcr itself: XOO-fS-f- und to be hus 999= expresses I ay, tlioro- tov M-liich, nines are digits, we fe remain- ■emainder elf, or out ning can himself )ught : — 3W much eypress- y means ion ma;i yinj>- tli<» product, will Iftave the same remainder, as if the ninps were east from the product of the factors," — provided the multiplication has boon rightly performed. To bhuw tliis, set down the quantities, and take away the nines, as before. Let the factors be 573x464. Casting the nines from 5-J-74-3 (which we have just seen is the same as casting the nines from 573), wo obtain 6 as remainder. Casting the nines from 4-f-G-j-4, we get 5 as remaiiuler. Multiplying 6 and 5 we o btain 30 as product ; which, being equal to 3x10=3x94-1=3x0-4-3, will, when the nines are taken away, give 3 as remainder. Wii can show that 3 will be the remainder, also, if we cast the nines from the product of the factors ; — which ia clFected by sotting down this product ; and taking, in suc- ecssion, quantities that are equal to it — as follows, 573x404 (the product of the factor8)= SxBO+T xlO+S X 4 x 1004-6 xl0-}-4= 5x99-fl-f7x9+l4-3 X 4 x 994-14-6 x94-l4-4= » 5x994-54-7x94-7+3 X 4 x 994-44-6 x94-6-|-4. 5x09, as we have seen [55], expresses a number of nines; it will continue to do so, when multiplied by all the quan- tiiies under the second vinculum, and is, therefore, to be cast out; and, for the same reason, 7x9. 4x99 expresses a number of nines ; it will continue to do so when multiplied by the quantities under the first vinculum, and is, therefore, to be cast out; and, for the same reason, 6x9. There will then be left, only 54-74-3 X4-1-64-4, — from which the nines are still to be (Tast out, the remainders to be multiplied together, and the nines to bo cast from their product ; — but we have done all this already, and obtained 3, as the remainder. EXEnCISES FOR THE PVPII.. Multiply By (37) 765 ^ 765 (38) 732 456 (39) 997 845 (40) 767 347 I'jtroducts * Multiply By (41) 657 789 (42) 456 791 (43) 767 789 (44) 745 741 rroducts 1 ■ 70 MULTIPLICATrON. .«n J' u! r "" ^yP^ors, or decimals in the multinli, cand, rnultrpher, or botli ; the same rules apply as when the niultipliar does not exceed 12 [43, &c.] (1) 4600 67 (2) 2784 620 (3) (4) 32-68 7856 26- 0-32 (5) (6) 87-96 482000 220- 0-37 2G2200 1726080 849-68 2618-92 19351-2 178340~" Contractions in Multiplication. r-.,f;ioT^^° '!i'' °«* necessary to have as many deci- aTd\&"lier-' ^"'"'*' ^^ "^ ^" ^^^^^ -^^^^P^--^^ un?erM;r^r'''? *i' multiplier, putting its xmlis^ place oZ? I' 1^ ""^i ^^, *^^^* denomination in the multipli- cand, which IS the lowest of the required product. ^ ^-.^'iW ^^.^^«h digit of the multiplier, beginninff With the denomination over it in the multipl cand ; Tuf addmg wha would have been obtained, on multip yW the precedmg digit of the multiplicand-unity, if^the nmnber obtained would be between 5 and 15 - 2 if between 15 and 25 ; 3, if between 25 and 35 ; &c frt^Kl }7^^^ ^T^'''''^'''''^ ^^ *^^« products, aripinff from the diflFerent digits of the multiplicand, stand ia the same vertical column. ' Add up all the products for the total product; from which cut off the reqmred number of decimal places. 59. Example 1.— Multiply 5G784 bv 97324 sn oa +« have four decimals in the priuct ^^-^^^24, so as to Short Method. Ordinary M.thod. 56784 42379 511056 39749 1703 113 22 55^2643 5-67r t 9-7324 22i7136 1131568 1703i.")2 39748 8 51105G '4 55 -2044 601 (] i j^LM. MULTIPLICATION. 71 9 in the multiplier, expresses units ; it is therefore put ander tho/o«r</i decimal pliioo ol'tho multiplicand— that being tho place of the lowest decimal required m the product. In multiplying by each succeeding digit of tho multiplier, we neglect an additional digit of the multiplicand; because, as tlie multiplier decreases, the number multiplied must in- crease—to keep the lowest denomination of the ditterent pro- ducts, tho same as the lowest denomination required in tlie total product. In the example given, 7 (the second digit of the multiplier) multiplied by 8 (the second digit of the mul- tiplicand), will evidently produce the same denomination as 9 (one denomination higlier than tbo 7), nniltiplied by 4 (one denomination lower than the 8). Were we to multiply tho lowest denomination of the multiplicand by 7, we should get [4(5] a result in iha Jift/i place to the right of the decanal point ; which is a denomination supposed to bo, in the present in- stance, too inconsiderable for notice— since we are to havo only four decimals in the product. But we add unity for evt'.ry ten that would arise, from the multipl cation of an ad(". tional digit of the multiplicand ; since every such ten consti- tutes one, in tlie lowest denomination of the required product. When the multiplication of an additional digit of the inulti- pliciind would give more than 5, ..ud less than 15 ; it is nearer to the truth, to suppose we have 10, than either 0, or 20 ; and thereiore it is more correct ta^add 1, than either 0, or 2 When It would give more than 15, and less than 25, it is nearer to the truth to suppose we havo 20, than either 10, or SO ; and, therefore it is more correct to add 2, than 1, or 3; &c Wa may consider 5 either as 0, or 10 ; 15 eil/ier as 10, or 20 ; &c. On inspecting the re.sults obtained by the abridged, find ordinary methods, the difference is perceived to bo inconsiderable. When greater accuracy is desired, wo should proceed, as if we intended to havo more decimals in tho product, and afterwards reject those which are unnecessary. EvAMPLE 2.— Multiply 87653^ by -5704, so as to hav» 6 decimal places. Mt 8-76532 ^ 4G75 4383 G13 62 3 6051 ,s^i lil'ri :_. ai 31! :U 72 MULTIPLICATION. Tliere arc no units in tho multipljpr; but, as the rule dirocts, wo put its units' place under tli^ third decimal place ol the nmltipHcjuii. In multipivlng by 4, since there is no di;,'it over it iu Lho multiplicand, we merely set down what would have resulted from multiplying tho precodiu«r dono- mmatlon of the multiplicand. ° ^ a Example 3.--M.dtiply -4737 by -6731 so as to have docnnal places in tiio product. •47370 137G 284220 33159 1421 47 •318847 _ Na have put ^.he units' place of the multiplier under tho suth decimal place of tho multiplicand, adding a cypher, or su^iposing it to be added. Example 4.— Multiply 84G732 by -0050, sc as to have lour decimal places. , 84- 0732 G5 4234 508 •4742 Example S.—Multiply -23257 by -243, so as to have four decimal places. 23257 342 465 93 •05G5 AVe are obliged to place a cypher in the product, to mako up the required number of decimals, 00. To multiply by a Composite Number — KuLis.— Multiply, successively, by its factors. I 8 the rule inial ^lacu tlioro i- no ovnx what iiug dcno- to have 6 [inder the sypher, or } to have liave four to malcG i MULTIPLl ATION. 73 EyAMPT.K— Multiply 732 by 90. 90=8x12' Lhotoforo 732x1"' 732 x8x 12. 732 8 [35 I 5850, product by 8. 12 70272, pi .act by 8x12, or 90. If we multiply by 8 only, Ave multiply by a quantity 12 tinios too Hinall ; ami, therefore, tlie product will bo 12 times loHH than it slioul ' We rectify tliis, by making the product J J times greater— ...at ia, we multiply it by 12. fU. When the multiplior is not exactly a Composltcf Nui!i))or — lluLi:. — Multiply by the factors of the nearest com- posite ; and add to, or subtract from the last product, >s{) many times tlie multiplic \ as the assumed compo- site is le.ss or greater than the given multi2)licr Example 1. — Multiply 927 by ^7. 87 = 7 X 124-3 ; therefore 927 X 87 = 927 X 7x 12+15 = 927x7x12 + 927x3. [34]. 927 7 0489: 12 :927 X 7. 77808 = 027x7x12. 2781 = 927x3. 80049 = 927 X 7 x 12 + 927 X 3, or 927 x 87. If we multiply only by 84 (7 X 12), we take the number to bo multiplied o times less than we ought ; this is rectified, by adding 3 times the multiplicand. ExAMPLK 2.— Multiply 432 b y 79. 79 = 81-2=9 x 9-2; thoroforo 432 X 79=432 x 9 X 9-2=432 x 9 x9-432x 2! 432 9 3888 = 432x9. 9 34992=432x9x9. 804=432x2. 34128=432 x 9 x 9-432 x 2, or 432 x 79. IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I .25 am 2.2 1^ illlU Ik ..ill IIIIIM u mil 1.6 ^. % <9 /^ * 'c^l ^^ > ^ cf ^^#^5^' .** :>/' ^-^ HiotograDhic Scieices Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 iV ^^^ •1>^ :\ \ ..'" '^ '^> ^ /^ v. "^n- ■^^f^^ l?.r 74 MULTIPLICA7I0K. fli nifn K '^ r^° ^^ ^}'. *''° composite number, we have taken Jh7product ^ subtracting twice the multiplicand from 62. This method is particularly convenient, when the muKipher consists of nines. To Multiply by any Number of Nines,— liuLE.— Kemove the decimal point of the multipli, cand so many places to the right (by adding cyphers if necessary) as there are nines in the multiplier • and subtract the multiplicand from the result. ' .ExAMPLK.— Multiply 7347 by 999 ' ^ 7347 X 909 = 7347000-7347=7339053. We, in such a case, merely muliinlv hv the ««»* i.* ^, convenient composite number,'and "Sra^I the muHiputnJ :xr;je^Ju«tS;4r ^^^- ^^ *- ^^^en; thuftrh'a 7347x999=7347xi000-l=:7347000-7347=7339653. 63 We may sometin.cs abridge multiplication hv oonsidermg a part or parts of the multiplier as pro- duced Dy multiplication of one or more other parts. ExAMPLK -Multiply 57839208 by 62421648. The mnl. tipUer may be divided as follows :-0, 24, 216 and 48 24 = 6x4 216 = 24x9 48 = 24x2 57830268, iimltiplicand 62421648 , multiplier. ''S??oP3^ • = P''0(^"«t1>y 0(60000000). I.ih8142432 : : product by 24 (2400000^ 12493281888 : product by 216721600/ 2776284864 product by 48. ^" 3610422427073664 product by 62421048. ,1 ^i'lP^o^?"^^ by 6 Avhen multiplied by 4 will dve the nrn trn\ Y 24; the product by 24, Multiplied ^9, iill eive^th; product by 216-and, multiplied by 2. tlie prodactTy 4!! fJt' Ji-'-f ''''.'\^^ !^^ difficulty in finding the places of the first digits the difi-erent products. For when thoro are neither cyphers nor decimals in the multiplicand— ndthorTiS n;ultiplication, we may suppose that there are neithei [4b, &c.]— the lowest douomination of each pro- MULTII'LICATION. 75 duct, will be the same as the lowest denomination of ths multiplier that produced it ; — thus 12 units multiplied by 4 units will give 48 units ; 14 units m.ultiplied by 4 tens will give 56 tens ; 124 units midtiplied by 35 units will be 4340 units, &c. ; and, therefore, the beginning of each product — if a significant figure — must stand under the lowest digit of the multiplier from which it arises. When the process is finished, cyphers or decimals, if necessary, may be added, according to the rules already given. -^ The vertical dotted lines show that the places of the lowest digits of the respective multipliers, or those parts into which the whole multiplier has been divided, and the lowest digits of their resulting products are — as they ought to be — of the same denomination. 48 being of the denomination units, when multiplied into 8 units, will produce units; the first digit, therefoi'e, of the product by 45 is in the units' place. 216, being of the deno- mination himdreds when multiplied into units will give hun- dreds ; hence the first digit of the product by 216 will be in the hundreds' place, &c. The parts into which the multi- plier is divided are, in reality, COOOOOOO 2400000 • 21600 48 .=62421648, the whole multiplier. We shall give other contractions in multiplication hereafter, at the proper time. EXERCISES. 45. 745X456^:=339720. 46. 476X767=365092. 47. 345X579=199765 48. 476X479=228004. 49. 897X979=878163. 60. 4 •59X706=3235 -95. 61. 767X407=312169. 52. -467 X- 606= '276942. 53. 700X810=567000. 54. 670X910=009700. 55. 910X870=791700. 66. 5001-4x70=350098. 57. 64 -001X40=2560 -04. 68. 91009X79=7189711. 59. 40170X80=3213600. 60. 707X604=427028. 61. 777 X •407=318-239. 62. 7407X4404=32620428. 63. 6767X1307=7537469. 64. 67 •74X -1706=11 -556444 65. 4567X2002=9143134. 66. 7-767x301-2=2339-4204 67. 9600X7100=68160000. 68. 7800X9100=70980000. 69. 6700X6700=44890000. 70. 5000X7600^--38000000. 71. 70814x90l07=63808-37098. 72. 97001X70706=7440658706. 73. 98400X07407=6295813800. 74. -56007x45070=25242-35490 70 WULTIPLrCATION. aolmtir"^''""'"^' '" ^1395; a pound being 7b. In 2480 pence how many farthings : four far- thmgsbemgaponny.. ° A«. 9920. I. J f;r'87lhunSf: p ■="' " ^■''"'"^' ''- ™r "f;,^; 'TO IT 1° .1, Ans. 1479. ton f "''' ' ^^^ ^^"^ ^^ ^^***^^ ^°«<^ ''^ ^25 a 'TO 71? 1 /. 7l7w, 012;') will 119 n^ ^T f /°^ *^^°S cost 4 pence, how much 80 llZ " '"'* • • ^^"- 448 pence. 80 Row njany pence in 100 pieces of coin, each of which 18 worth 57 pence ? a,,, r'inn CI TT^,„ f^^^^ jins. 5700 pence. t«inL RQ ^^^ypUons in 264 hogsheads, each con- taining 63 gallons .? ° L,, Tp,'" 82. If the interest of ^1 be £0-05, how much wi5i be the intarest of ^6376 ? ' Ans £l 8 8 cosll" "^ ''''^ """^'"^^ '°'* ^^'^^' ""^^^ ^^^^ 973 such 84. It has been computed that tho gold, silver and brass expended in building the temple%f So omon at Jerusalem amounted in value to ^£6904822500 of our money ; how many pence are there in this sum oZ Ts'Stir"''- . 1 ^-165VV57SC 85 I he followmg are the lengths of a degree of the ?em •1?):86'6'- 't"?"-^ ^"""^'^ 60480-2\ThoLin . n ' ,^^^^^^ m I»<iia ; 60759-4 in France • fiOS^fi-fi m England; and 60952-4 in Lapland. 6 feet beln' a st^Yt'^ r"^ ^''' ^^ ^^^^ «f ^'- above. P 1?.; 362b8l2 m Peru; 362919-6 in India; 364556-4 in France ;r^650_l 9-6 in England ; and 365714-4 in Lapland 86 The width of the Menai bridge between ?he points of suspension is 560 feet ; and th? weigtrbeLL these two points 489 tons. 12 inches bein/a foot and 2240 pounds a ton, how many inches in°the fomTr and pounds m the latter ? loimer, 87 Th..n ^"1' ^^2^.i?«^es, and 1095360 pounds. 87 There are two minims to a semibreve • two crotchets to a minim ; two quavers to a ciSeV- wo semiquavers to a quaver : and two demi-sem quavcrslo I rerSre^P ^^^ demi-semiquavers i ~^ Ans. 221 M' M ULTI PLICA n ON. 77 88. 32,000 seeds have been counted in a single poppy • how many would be found in 297 of these ? Ans. y50400o! 89 9,344,000 eggs have been found in a single cod Lsh J how many would there be in 35 such ? ^^ „„ , Ans. 327040000. ^ 65 When the pupil is ftimiliar with multiplication, m workmg, for instance, the following example, 897351, multiplicand. 4, multiplier. 3589404, product. He should say :— 4 (the product of 4 and 1), 20 (the pro- duct of 4 and 5), 14 (the product of 4 and 3 plus 2, to be earned), 29, 38, 35; at the same time putting down the units, and carryin^; the tens of each. QUESTIONS TO BE ANSWERED BY THE PUPIL. 1. What is multiplication .? [24]. 2. What are the multiplicand, multiplier, and nro- duct.? [24]. ^ ' ^ 3. What are factors, and submultiples } [24] . ^ 4. What is tlie difference between prime and compo-- site numbers [25] ; and between those which are prime and those which are composite to each other ? [27] . 5. What is the measure, aliquot part, or submultiplo of a quantity ? [26] . 6. What is a multiple .' [29]. 7. What is a common measure ? [27T . 8. What is meant by the greatest common measure > [28] . 9. What is a c<9wmo% multiple .? [30]. 10. What is meant by the kast common multinle > [30]. ^ • 11. What are equimultiples .? [31]. 12. Does the use of the multiplication table prevent multiplication from being a species of addition .? [33]. 13. Who first constructed this table ? [33]. 14. What is the sign used for multiplication ? [34]. 15. How are quantities under the vinculum affecteJ!* by the sign of multiplication .? [34] . 16. Show that quantities connected by the sign o/ multiplication may be read in any order ? [35] . WL^ai 78 iiirisioN. ,0*';^',"'^ ""■■ '""llilJiw exceeds 12 > ran ' cccoAs I2t [Lt "'''' "'"" °»'^ *» nluUiplicHud *; J°' '^^^^''"■« tl"= rules when the mnltipUcand mnl ' a^'^T-r; ""''.?""" "W'"''-^' ^ decimals ?[43;&Ti: 2i' F."^i' °'""'P"''f<'? P^o^^i •' [42 and 53]. wtdgn?.Sl' ITsr""^'' ""^°"' ^"pp-»g » n»nfbo?„7del:rpLt¥^r5sV' '" ''"^ " '"^--^ ^ J8. How may we multiply by any number of -Jnes > [esf: ^""^ '"" °"'"'P'i''^«en «ry briefly performed ? SIMPLE DIVISION. orde'eminatl""""'' "^ ^PJ*""-*' ^■" "»'- o^"^ calMTlTJ?'^""'™ '" *■"> °"' ''"w oft™ one number called the divtsor, is mnlaineil in, or can *, /^/J;,T ' another, termed the divi,i,^,l ■ .1 ^ i ™'«"/™« /-».//» is call d tt S;„T DKr™'f^P''"^T« u« t» tell, if a quantity be dWded into a cL?" '""f ' "'Xi^^'d^"-"' ^"^ "^ ti'lCt^ofrh" """'" wnen the divisor is not contained in the divid^n.? p.-ocea, would be required to'dLTrer^SyTub* vivitioa. 70 trading it~liow ofton 7 is contairiod in 8063495724 jv^i.lo, a.s wo_«h.ll liud, the same thing can bo effoctcd by dicmmi, m less than a minute. 08. Division is expressed by -f-, pLvced between the .v.dend and divisor; or by j.utting^he divisor und J the dividend, with a separating line between :— thus 64-3=2, or-=.2 (road (3 divided by 3 is equal to 2) means, that if 6 is divided by 3, the quotient will be 2. <)9. When a quantity under the vinculum is to ).c divided, we must, on removing the vinculum, put the divisor under each of the terms connected by the .i^^u of addition, or subtraction, otherwise the value of wliat was to be divided will bo changed ;— thus 5T6^-^3=:r- 6 7 ' ~ ---!.____. f^^. ^^j j^^ j^^^ ^^.^..^^^^ ^1^^ ^j^^^^ ^^^^^^^ vvc divide all its parts. _ The line placed between the dividend and divisor occa- Monally assumes tlie place of a vinculum ; and there- ore, when the quantity to be divided is subtractive, it \uli sometimes be necessary to change ^' - * already directed [16]:— thus --+ ^^""^ the signs— as 6 + 13—3: but 27 3 ^5— 6 + 9 ^ 27—15 + 6- 3 2 -9 2 For when, aa in these cases, «// the terms are put under the vinculum "thf : '' v^f '^^? '"^^^^^*^''^ ^^S"^ -^'^ concern S- IS the same as if the vinculum were iSmoved alto-ether • and then the signs should be changed i../; .^a tJ what they must be considered to have been S tl e vmculum was alfiv^ed [16]. -^ ^ When quantities connected by the sjcrn of multinlien tion are to be divided, dividing Iny one° of trS.s' .nil be the_same as dividing the product ; thus, 5X10 X 2o+5= - X lOX 25 ; for each is equal to 250. To Divide QuanlUits. 70. When the divisor does not exceed 12, nor the dividend 12 times the divisor ' 80 DIVISIOIf. B.;le.~I imd by tlio mnltiplicntion taMo tho groutost number which, multiplied by the divisor wlU give a product that doo.s not exceed L diviS' 7\^ will be the quotient required ' number td'fh/r^" '^'' t^^''f *^« P^«^^«* «^ ^^"'^ anv w tlw i- • ''^''"' ', ''^'"« '^'^^^^ *^^" remainder, if any, with the divisor uuder it, and a lino between them aro^S T '''^ ^^-/""Itiplication tablollllo times G The total quotient is9+|,or9^; that is, ^=9^ ^ji :^:S^t^XS^;rt^"^or, w^ean^^ct it ber'of^i^rThrcivTsox-Lnt;\i^ *?« f-*-* --- is, tlie greatest mull pie of 6 w], oh •i^'''"V'^^ dividend; that ber to°be divided The mnlTinH^^^ not exceed tho num- ducts of any two m^mhL '^ '^"^ ^^^'^^^ ''^'"^^■^ the pro- therefore iJirlblesu" to obt^i"^^!^^ ' "'^' ''''''^' ^^' '^"'^ must not exceed the dhiSend"^^ ^''^^^"^■«' ^•"'* leave a number equal to or '-olf' T° «".''t^''^«ted from it. hardly necessary loremrk t at .7 * i'""' '^'' ^^^^'•^«^- ^^ is been subtracter! .4 ofTon 1.' '.t -k'*" ?"'''"'' ^^"^'^ "«t have number equal to or ffreatel tlfan u'' ^''? i''« ^^'^^'^^"^ ^^ ^ quotient answer tleon^^Hnn / '^ r '''■'' ^'^^^ ' "'^^ ^^0^1'^ ti.e taken from the dividend °' ''"' "^'"'^ ^^'^ '^^^^^^^^ ««^^IJ bo anf^SenTf(4;r'JreSrnVL^ T'^^-V ^^ ^'^ ^^-«- remainder, what it i When^j. Je H';^ *'-"i'' ^' '^"^ J-eahty suppose tho dividend div eel iito IT'^^'f'' ""' '" these IS equal to the product nfThi- "^^ P'''^"*^ ^ «"« f>t' pie given, f =:^i±4=,l4 4 4 72. When the divLsor does not exceed 12 bui M,« dividend exceeds 12 times the divisor-- ' '' DIVI.SION. tablo tlio li visor, will idend: this luct of this mainder, if yeen them. 1 58; or, in )y G. 10 times G Teforo, doea ible, that 9 tly 6 is con- 9 quotient; given num- 4 an effect it ateat num- dend; that I tho mini- 's the pro- Is 12; iitid luire; this id from it, sor. It is 1 not liave idend if a would tlie ■ could bo le divisor be any er, we in s ; one of lent— and i between ^s, hy the lie cxtiia,- bui the 81 liiiHBssias r 10 ino mg remainder, when there is one ind de /r/ '^I contain the divisor consido,. .V + ' ^"r"^"^) ^^^«« "ot the next lower and ^d, I ^ '''''' ^^ "'^'"^ ^^' the dociuml point. ' ^^'''' removed from "-Jis^et;:rit,"rit!t:; ^'^^ ^t^^-^ *'- [70]-,vith the div sor u nd l 7 i ^'^''"''^-^ ^^'""'^^'^^ l'otv.oen tliem ; o ^writ nt tl!' d" ' f'^"'-"'"^^' ^^"« quotient, proceU with t o^di! ""i^ ^^''"* ^" *''^ vonKundor'ten 1?^! t t nt^^'th'e ttt'l""'"' r'' mmathm ; proceed tliua Z^7fh • ^ ^''''''^'' ^*-^"'^- nntil it is so tr flin .7h. T- "'? '' "^ remainder, or iaconvenfence ° '^"^ ^' "'^^Slected without 73. Ex.MrLE.--What is the quotient of 04450-^7 ^ Divisor 7)G445G dividend. ' ' 1)208 quotient. i« greater thin GO t mm ?1 '' •'^'' '1* '^^^'"'''"'^^' ^^'^'^^''^ to he nut mle J T: \^'T '"' therefore, no di^it o;>evei, put a cyp icr m that place, since no digit B2 DivisroN. l! drnfli4 '^ ■ • HI11L3 ^ JiiinUrods nvo (iT.ir.n,r i i i the go tens, which tens divid( in« phice of the quotioi.L, juiu cue quotient is fu,i„d to bo 9208 eaetly ; H.at is, tt^^ 92O8. b/fi J ^""'"'^ 2.-What is tl,e quotient of 72208, divided 0)73208 proceed with the division as followa-- ^ '' "'"-^ 0)73208 ' ■ ., . , i22rf333,&c. L-onsiderinoj the 2 units inPt- f„ xi dend, as 20 Tenths, rpel-ceivetZ a" ".f ' '^ ^''^ ^^^^^- three tenths times and I^.tn if •''''" «'' '"t" t^^o'^ 6 (=0 times 3 emhTf^t^f ' I ;f^^ ^ ^^"^^'^ ^^'''^ into .cThund^edths Vl:^^^ ^^-^ wn;^^ Ilin dooiiutil euMo, i»ro(Iiu!n of tlK.usatiil.s ^iiiu.l» iilrciidy '• "so" into usiind times; I is Icsa tliau oos not Icavo it is not too I)laco (if tlio ng adcloj to •^ thdiisaiul) iivo 14 Imii- il.s, and Icavo 3tly 14 Jmn- da' ])Iai!o of 7 will not Jnes 7 are 7 Ji'in;; tho 5 units of tho tiniOH, loav- in the tens' urea fiii'tJior ni nation of phor. Tho !»o required -=9208. 28, divided ■ after tho >i" wo may ' Hio divi- into thoui itlis times t <> in the onths ro- (i will go e 2 lum- DIVISION. 83 denominations of f]/c qu.)tient; wo m- v^ 1 o ■ ""'r''^'« put dowa m the quotient us nutn^ S^. s w ^.'^.'7^ liual remainder so small, that it mtiy be m'glecTid. ' 75. Example 3.— Divide 473G5 by 12 12)47305 ' * 3U47-08, &o. In thia oxampie, tho one unit loft Caftcr obtaininn- fKn 7 • ntr'i-^ tr ^'r. -r^-^i as io to^rsrit StZths'';;;;x e^^^^^^ "^^^^'"« to bo set down ; ine renins place ol tho quotient— except a CYphcr to kr.pr. tho following digits in their proper places 'Fm in ? !i ^ are by consequJnco to bo coLiXro. ^^^ 12 will go int^ 100 l^undr^dZ 8 idr tl '^^ toliSuYtt iSn^^ ''' '-' -^« m whTn^rdosiro ' Example.— Divide 8 by 5. «-^5 = l'Sorl-37, &o. 76 When tho pupil fully understands tho real deno n^inations of the dividend and quotient, he may proceed for example, with the following "^ pioceeu, 5)40325 In this manner :— 5 will not go into 4 5 I'nfn da a r find 1 over Cthe 40 bpi'nn. r>f Jv i • . ^"' ^ *i»ios produced it) 5Tnto 13%w- ^"' the den<imination which times and 2!,ver' 5 in l oJTtir:^ ^ r^' ^ ?"^° ^2, 6 ' ""^^^ ^"^ ^^^ remainder. -ligil. of the quotL^aro a/eo Str'lV' '' Ti"""" »'' «>« .ub.™t.. rro„ t.4^"^nf istsi;™ :i,'ir.ii' riL° 84 nivisiiotv. Thni, }f T) gooji tlio quotient (544.k5l7 w« '?„ ^,/^' ';• '", '""'"'f?' ["•' «xa»iplo, cnvi.Iond Huitc 1 to M, roCs"'? , '• "• '^'^'''••••^»T. to render tl.o >vlule, at tI.o «amo t me/wo e vo i s v^h'"' '''"■'•'"•• i'''^ '"'"•'"• coruort • *^" ^*-''^^ "** value unohangod; It bo- Thousands. Hnr„lreJ«. Tuns ' ir •, E«wh j.art being divided hv 7 f l,n hr ^'^' (^^^^f^^)' dividend, with tifeir ^iotiv'o q'uotonln^'^Jirbr''^^'^^ "' ^'^^^ and the nuestion is Zf./v *: '^ quotient in a lower; dividend~it3di?Lint de onf/n'V '" i''''''' ^''^ «" '"to tl'O venicnt way \Vo can n T^ T ^"'"» taken in a,;y con- Hhall have t^o add L t e LerdLonX.';;'- "^ '' 'I'' '''^' ^^° With the higher. '"^ ^''^^^^ denominations, unlesa we begin th^x^n5;Lf"r;f S divide:;!; 'Y r''^?' - p^* "-^o- it belong,, to that demmdnatiZ "!;';'''' ^''''^'''''\ '^' ^««'^»«« of time "(indicated by S of'thn. expresses wliat rumber tan bo taken from the coSInn'^"'^ "lenonunation) the <livi.sop thus tlie tons of the onot^^n?^ "" T"'^ ""^ *''« 'Jividend:- the divisor can be t ken fro^lT'''^; liow many tens of times hnndreds of ti^e mio ient tZ "" *,'"' ?^ *''« dividend; tho 6e taken from tlie TuXds &c '"^ '''''^'''^' '^ '''^'' ''^ '^^ m^:ih::;it^;;j;-^i;So7^;,- l^f belong, to the total re- lower denomination t w i tti ' m •* "'i-^''" considered as of a He Asorv OK V -Ve a Jo f o 1 ''l"""- ^"'':^"^*^^ ^" *''« quotient, the highest deLm nation canablo n^ ■^'' ^"''"^'""^^^^ ^« «^ tliougli it may not cont^i n E J?„?^ ^^^"^^ i^ quotient; and press«. by a di<r it of one don., • I-'"'" 5' '''''^^^'' ''^ times ex- Lnbc., ot^im J^/p^oTed'ryTn^Ua^ «^"^'- ^^ -- t2!J:reSnXrtheS[t^5eS^S'"^ oach.produot, is the *) mucjj of it as is neoessirv f^ '/"'^ ^^ ^^''"3 down" only iooking for a d L t ?n ?£ JmnX';;' Pr''"^°^J^°*- '-Th^^' in ^m not be ncc^essary tota^e Sy;^^^^^ quotient, it Of the dividend ; since hev canm f « n .""?, *^'' *'"''' °^ ""^ts <lre.s of times the aivi.iJ?„^;rfaS f^oI^thrriS.'^'^* DIVISION. 85 Acyvher mnst bo n.Mc.l [Sec. I. 28], when It ia rcquiml tlio orwe. except it oinoM botwcou thoiu un.l me .leciuml p-int. Hr:AsoN <.K VI.-Wo .....y continuo ti.o process of division. If wo plouso, a.s o„ff as it ih pusible to obtain .,uotio»ts of ani icnon.m.uoa. Q.iotients will l.o po<luce.l although th ore are nu lung.r any «.gn.nount figures L the dividend, to which wo can add tho succcHHivo roniaindora. 78. Thfi BiiialW' the divisor the larger tho (motiont- lor, the smaller the parts of a given quantity, tho groat-r tlum- nuuiber will bo ; bt.t is tho least po.ssiblo tlivi- sor, and therefore any quantity divided ))y will <rivc tho larg(.st possible (^uoticnt-which in infinity. "Hence though atiy quantity multiplied ' by is equal to 0, any number divided by is e.pia! to an infinite number. It appears strange, but yet it is true, that-=:- ; for each is equal to tho gre.nlr.st pns.siblo number, and one, thcrefc 3, cannot be greater than another-the appa- rent contradiction arises from our being unable to form a true conception of an infinll.c (luantity. It is neces.^ary n.-m^ •^•'Tf '^'*\f'^"^ ^ ''' ^^''' ^'^«^^ "^dicates i quj tKy^inhnitely small, rather than absolutely nothin^r 7J lo 2>roi-c Dlcis}^n.~lSlnmiy\y the quotient fv the divisor ; the product should bo equal to the divi' Uend, minus tho remainder, if there is one I'or, tlio dividcn.I, exclusive of t!ic ronaiuder, contains tho Ir'^t' rdivl.'": '• ""rr '""''r'''' '^^ ^^c quotient ";?.«! eoml nf r ',''i^''^'^'" ^'"'^ ""'"^'^'r of times, a quantity U i 1. tvH tirirnTr • "",'"' '''« r-naindcr, will be p^Uucelf it luiiows, tiiat adding tho remainder to tlie product of the divisor and quotient ahould ^ve the dividend. ^ ^ EXAMTLK 1.- T708 n i.t i. 'L'^^32 ,_ rrovo that -— ^-=1708 4 PuooF. 1708, quotient. 4, diviKur. sov and quotient, equal to the^dividcrlf'^' ^^'"'^"'^ "^' '-^'^■" ExARipi.E 2.-.l>r<,vo tliat ~"^^^- ioo-m ^ Proof. ' , ' rnooF. or 122^4 122^1 7 **^^*^^=='iivMcu>l "limn:., tliediuniii V- 7 i: 2 86 DIVISION. 2)78345 EXERCISES. . <2) 8)91234 (3) 3)67869 (4) 9)71234 4)96707 6)970763 (6) 10)134667 12)876967 (7) 6)767456 7)891023 (8) 11)37087 (12) 9)763457 SO. iVhe>i the dividend, divisor nr hnth w hen the dividend contains cyphers— 68 7 times, it will bo -ntrt fiRrin / -"^P^^^' if 8 w^l go into than 66) 100 tiSS ^orftLtTtiL^s^ir^oV.TJr-^ ^•'^^^*- i^AMPLE l.-What; is the quotient of 568000^4 i 4 --14 J, therefore — —-. = 142000. E^mPLE 2.-What IB the q, ti^nt of 40G0000-.5 ? *; -SI 2, therefore —y— =812000 [Sec. L 39.]. 81. When the divisor contains cyphers— can be taken fr- m it 100 tfmes^le^rSen. ''' ^^'^ '""^' ^ ExAMPi.E.-.Wliat is the quotient of -^ I ro °00 - - 58 ■g=/ ; therefore ~-= 07. 800" DIVISION. 87 (4) 171234 (B) 37087 12) 33457 ■ contairi re apjtli, them. removo 2rc liavo ) be tlie e Uiviaor ' go into d greater ; 2 39.]. we the cyphers times it '; 6 can times 6 82. If both dindenrl and divisor contain cyphers— RuLE.-DA'id^ as if there were none, and move the quotient a number of places equal to 'the dfiL'enoe t^Hr'''-?^""^?'' '^- ^^P^^^*^ "^ *he two given quan! t ties r-if the cyphers in the dividend excefd thole hi the dmsor, move to the left; if the cyphers i^ the divisor exceed those in the dividend, move^to Te T^ght ExAMPLEa. (i) 7)63 9 (2) 7)6300 (3) 70)63 " 0-9 70)6300 700)630 (6) 700)6300 eoo 0-9 —90 —0^ 9 of Pvnit^l^- ^^^"WK the difference between the numbers 83. If there are decimals in the dividend— KuLE.— Divide as if there were none, and move the quotient so many places to the right as there are deoi- The smaller the dividend, the less the quotient. ii^XAMPLE.— What is the quotient of -048-5-8 ^ 48 -048 g-— 0, therefore -^==-006. 84. If there are decimals in the divisor— KuLE.— Divide as if there were none, and move the quotient so many places to the left as there are deci! The smaller the divisor, the greater the.quotient. JiXAMPLE.—What is the quotient of 54-i--006 ' 54 54 ' ■ g-=9, thererore;^==9000. visor-"^^ *^'''^ """^ '^''*'"'^^' '"^ ' '*^* ^^^^^^^^ «n*^ <Ji- Kui.E.— Divide as if there were none, and move the quotient a number of places equal to the difference M 68 DIVISION. between the numbers of deeiraab in the two given quan- t.t.e« :_.f the deciraab in the dividend exeeed thor?n the divisor, move to the right : if the decimals in tim divisor exceed those in the dividend, move to the left Examples. (1) 6)45 9 (2) 6) '45 •09 (3) •05)45 900 (4) •5) -045 •09 (5) •005J_450 90000 (6) •05) -45 9-00 ■ *],f^^''^;'~~?''^'^^ ^^ ^ *^^^« ^«r« i^either, and move the quotient a number of places to the left, equdlo the number of both cyphers and decimals. ^ Example.— What is the quotient of 270-f--03 ; -^=9, therefore, 270^-03=9000. in ?he d^koTl'" ^'""^'^' " '^' ^^^^^^^°^' -^ «^^-« Rule -Divide as if there were neither, and move the quotient a number of places to the ri4t eauS to the number of both cyphers and decimals. " ^ ^Zi^^Z^S^^'"'^' '^' «- «^P^-- - the E.YAMPLE.~Whaf ia the quotient of -18^20 ? -^ = 9, therefo-e ■l| = 009. 20 The rules which relate to the management of cyphers and decimals, m multiplication and iS division-/hou4 iumerous--will be very easily remembered, if the pupil . " ■' — """^ -^d^fcc. tu yu ine cuesi v) ciihor ;iven quan- id those in nals in thu the left. I move the ivisor move ogether, the eir separate (6) '05) -45 9-00 and deci- md move equal to a la in the Icyph ers ad move equal to ra in the cyphers -though he pupil iihtir (13) 8)10000 (14) 11)10000 DIVISION. EXERCISKS. (15) 3)70170 (16) 6)68630 89 (17) 20)36623 (18) 3000)47865 (1^) 40)56020 (20) 80)75686 (21) 12)63-076 (22) 10) -08766 (23) •07)64268 (24) •09)57-368 (25) •0005)60300 (26) 700) -03576 (27) •008)57-362 (28) 400)63700 (29) 110)97-634 88. When the divisor exceeds 12 . The process used is called Ions; division • thif i<? wa perform the multiplications, subtlaclLrr&c , nJl and not, as before, merely in the mind. ' This will be imderstood better, by applying the method of longdivi! Tat r ihanTr^ '• ^" ^^^'^^^~'^-- divisor noticing gi ater tlian 12 — it is unnecessary. ° Short Division : 8)6763472 720134 the same by Long Division. 8)6763472(7204:]4 56 16 16 > 34 32 27 24 ~82 32 dil?renM,aTof X'f 'r^"^^;^!-^ *^« ^^ ^'J the uuiuent paits ot tJie quotient, and in eacli case ,ei \\mo,x 90 DIVISION the rroducf mLlmct it from «,« corresponding portion of the <1 vKlond, write tlie romaindor, and irinn ffJnth^^, quu-od digu, ,>f the dividend, Ali this 3 beTne when" dent' .StH^-i^te™'- *" '^'^ '''' "' ""^ *"" III. Fidd the smallest number of dibits at ih<^ 7pff and set down, underneath, the remainder, if there is any. Ihe digit by which we have multiplied the divisor IS to be placed in the quotient. ^ . T* J? *.^''' remainder just mentioned add, or, as it is said " bring down" so many of the next' digit or cyphers as the case may be) of the dividend, as are required to make a quantity not less than the divisor and for every digit or cypher of the dividend thus brought down ^c.^^ .,^, add a cypher after the digit last placed m the quotient. ^ VI. Find out, and set down in the Quotient iht> nnmUr of times the divisor is contained if^hTs qUn! tity ; and then subtract from the latter the product of tlie divisor and the digit of the quotient just set down. J^roceod A^.th the resulting remainder, and with all that succeed, as with the last. A- ^?'}l *^®/^ '^ "^ remainder, after the units of the dividend have been " brought down" and divided, either place It into the quotient with the divisor under it, and a separating Ime between them [70] ; or, putting the decimal point in the quotient-and adding to the re- mainder as many cyphers as will make it at least equal to the divisor, and to the quotient as many cyphers mi7ius 071^ as there have been cyphers added to the remamder— proceed with the division. DIVISION. portion of ^wi the re- done when lid be too the divi- e for the ■ the left quantity lem, the contain ; there is e divisor ', as it is igita (or , as are divisor ; Qd thus he digit mt, the s quan- duct of t down, all that of the , either it, and ing the the re- t equal lyphers to the t)0. ExAMPLK 1.— Divide 78325826 by 82. 91 82)78325826(955193 738 452 410 425 410 158 82 762 738 246 246 fnt?7« V '' -^1 ''l*^ ^ \ ^°' i"*^ ' ^ 5 ^»<^ it ^^i" go 9 times tV T^ ''i'',^^ P"* i" *^^e quotient. xvni L T-''^*,^*' ^^Shor denominations in the quotient if the DunTl ns hT '''' 'T'-'"^'' ^'^ P^-'^P«^ t« a«certo,in, ff titrtr!;h7ch th^e^roit" ^^^^^^^^^^^ '^'^'^ *^- -^--^ 6 thnes'"! t'"^^* ^.'^'"'..^^ ^^^« 4^- i^^to^vhich 83 ^e LiiB uivihoi tiom 4&J, which leaves 42 as remainrlpr 49 with 5, the next digit of the dividend, makes Sr^nowhith um me quotient. Ihe last remainder, 15, with 8 tliP n^vf digit of the dividend, makes 158, into whcr82 toi once^ leaving 76 as remainder:—! is to be nntln tK^ g?es once, rlToK^l ^ ^""Stit down expressed mut* t , ij Therefore 78325826 82" =955193. 02 DIVISION. Sample 2.-I)ivido G4212S4 by G42 042)(;42i284('T^( 02 042 ^ 1284 12S4 ing tr'thrncx?d?'^' r??, ^^^r^"^ ----^-. nrl,.. 1. TJh) next di^ of 't e 1 !n ' T-^'" ''' '^t'''^"^' ••^''^«^- *!'<^ «o digit in tJic^ not on h vhS "' '''' '^^'"'^ ^^''^^' g'^^'^ another cypher: and for sin l- r vf ' '^""^'^^l^^^tlj^ ^vo put down the next •X/ihn?''v ''."'' '*'"'^'^"- i» hi-bJivr gives^io romainderUve put 2 in'fT "'' f^'^''^"^' ''''''' ^"^ 91 Whm-. +1.nv^ • "^P"*^ J" <^^"c quotient. division, adding deoimaS SesfoTl '^''^ T'^^ ^""*^""« "'« o uLcimai places to the quotient, as follows— ExAMPM 3.— Divide 79G347 by 847. 847)79G347((i4010,'&c. 3404 3388 convenient to have two on nSS Zi ■ '""^^ *''* ^'^ '' ^"^^'^ !i;fl DIVISION. 93 6425x 54 ^± ViTsT&c. 102 64^ 485 432 53, &c Rkason of n.-This, also, is only a matter of convenience Rkason OF III.-A smaller part of the dividen-l woul "ivo no digit m the quotient, and a larger would give more than Keason of IV.-Since the numbers to be multiplied, and the products to be subtracted, arc considerable, it s not so convenient as m short division, to perform themultiplicXns and subtrac ions mentally. The i^ile directs us to set c bwu onlTLitfe" "" ^"^^■^^^^' '^^^"«« ''' ^^^*- - *^- -- liFAsoN OF V.-One digit of the dividend brought down would make the quantity to be divided one denomination lowe? than the preceding, and the resulting digit of the ouEt also one denomination lower. But if we are obliged tTbrfn - down two digits, the quantity to be divided is fwo denoiS? nations lower and consequently the resulting digit of tlie ot^o tient is ^^/»fl_ denominations lower than the preccdino-i-which uro-^a'c^vnliet^^'/.r*'*"" ^'^" '■ 281 is exp^ess'^St; using a cypher In the same way, bringing down three S't' '/ the dj^j^^^^^ ^^^j^^^^ ^^^ denomination thrle places lower^£n the hZ^ '' "f ''''''''''' ""'^^^ clenominSs lower tnan the last— two cyphers must then be used Thn T.^TT'^'V'''^^? ^"'^ an/number of characteis whether significant or otherwise, brou-ht down to any remainder J^^l^fl ""^ ^^.-^^« ^"bt^-"'^* the products of the different parts of the quotient and the divisor (those different Ss of are ?ounS th^„'/°^ ^^^ '^7'' «"«««««-ely accorSg^aB they are tound), that we may discover what the remainder is from w W J'A'''' *', "^Pf '* *^^ "^^* P^''*^*^" «f the quotient FroS no 3ecTmaYs?nT1? --'^ ^IP' '' ''■''''''''' *'^<^*' if there ar^ noaecimaism the divisor, the quotient fifmre will alwiv^ hi It is proper to put a dot over each dimt of tlie divi- dend, as we bring it down ; this will prevent our Ltei- tmg any one, or bringing it down twice. ^ 94. When there are cyphers, decimals, or both, tho 94 DlVIiJJOX. 9o. lo prove the Ihmion.—mxMMy the ouoticnt by the divisor ; tLe product .should be iL\ to the div - dcnd, mums the remainder, if there is anV '791 ^^.^io^Fovo It by tho method of "casting out tho KuLE.— Ciist tho nines out of the divisor, and tho quotient ; multiply the remai.ider.s, and cast tlirni it fiH„u tiieir product :-that which L now left oi^ to out of tlie dividend minus the remainder obkined from the process of division. «"ucu uom Example.— Prove that ~J!~= 1 181 .3 Con^-ider^daBa'^ ;, ,. ^ '^'*-^^776-2 = 037/4. To try if this bo true, Casting the nines from 1181, the remainder is 2. ), . .. ". " i^'Oin 54, „ jn 2x0 = tasting the nines from G3774, tho remainder is . .0 The two remainders are equal, both beino-Q- henco tl.n multiplication is to be presumed right, ancf «Zi Iv the process of division which suppos "s it. ^^'"'^^'Ititntly. The division involves an example of multiDlicatinn • sJn^.* the product of tlie divisor and q,loticnt oi^ghfto be cnuaMo the dividend minus the re.uainder [7'J]. lieLe in mot^nc? !idiri^^;^zirir^' - ^'^y e.pij.;rc£;nf EXKRCISiCS. (30) 24)7054 (31) 15)0783 318f3 (32) 10)5074 452^, (33) .* 17)4075 35410 275 (34) 18)7831 (35) 10)5977 435A (30) 21)G78r (38) 23)707500 3141^ (39) 390)5807 (37) 22)9707 323 443|i 33309^5 (40) , 1400)0707000 14-8897 4035-3425 ic quotient to the divi- g out; tho •, and tho tlip nines 1 ought to tho nines iued fioni 3-becomos truo, 2x0 = . henco the iso(;[uontly, On ; since i equal to a proving [54], we (33) '17)4075 "275 (37) 2)9707 'ml 000 . 335-3425 4 (41) 250)77670700 303424 •00y4 (44) 64-26 )123 •705 86 2 -2803" DIVISION. (42) 67-1^^42 •002 96 (48) •163 ) -8297 49 5-4232 ' (45) 14 -86 )269 -0625 18-75 (46) •0087 ) 655 150000 In example 40— and some of those which follow— after obtaming as many decimal places in the quS as I'o deemed necessary it wiU be more accurate to cons der t^ic haTf'o?ftT Z?"u *^f ^^\r^r («i^«e it is more than one halt of It), and add unity to the last digit of the quotient. CONTRACTIONS IN DIVISION. 96. We raay abbreviato the process of division when rflf f^%«^ajy decimals, by cutting off a digit to the right hand of the divisor, at each new di4 of the quotient; remembering to carry what would have been «n tTff th^ the mdtiplication of the figure neglected!!! unity if this multiplication would have produced more 25r&c' T59T ^^ ' ^ '^ ""'"' *^''' ^^' "' ^''' *^"" Example.— Divide 754-337385 by 61-347. fli oA?!n.TLT^'''^- Contracted M^od. 61-347)75^|33 7385(12-296 61-347)754-337385(12-296 C1347| 14086:7 12269 4 1817 33 I226j94 590J398 652 123 38 36 2755 8082 i|46730 61347 14086 12269 1817 1227 "590 552 li 37 DC I'lVlSiOX. ;i'u-ntly. tho portions of the dividcna from vMch th ^ T'n li-ivo been Hubtracted. "What hIiouI.1 i.uvn h •^ .^ '"''^ the multiplication of ti.e digi nSteHin^^^^^^ '■"'" 97. Wo may avaU oui-selves, in diviaion, of oonfrJ vances very similar to those 'used iu m;Uil>lS;,n To divide by a composite number— lluLE— Divide successively by its factors. Example. -Divide 98 by 49. 49=7x7 7)98 7)14 "2=98--7x7,or49. 98 If the divisor is not a composite number wp canno,as in multiplication, abbreLte' tC process oxcept It IS a quantity which is but little less^than a c^edTthe """^ f"^' *^" ^"^*^^"' - the _d Tn.^- A- ?''''"' i'^'"^"' «^^ ^i^'i^e the sum by the precedmg divisor. Proceed thus, adding to thTrcLi ' der in each case so many times 'the foregoing .uoHcn or i s'nffiT^f '^'^^^^ '^' Siven divisor untilV "xlc^, i?^. atcTlLT/l^P- ''""''^*^'^\*^ *^^ exact quotien IS obtained--the /c5^ divisor must be the given, and not the assumed one. The last remainder will be the ti ue tilVlHlON. fff. K.VAM,.r,K.~I)ivido l)87(5r.;342r, by 998 9870G;j.,425=t)S70(i342r)^10()u" i97r>.jr.l=ys7lTo33^2qp425^U)00 4.JU1=.IU752^75T4-1000. ■' 0-7..0U0=4x2-|:701>]00() . 0()l4)4()=-7x2.flT^1000 OU00..420=:.ur^2:p4_^100(). 00004,,0208=aT^2+^^908 icnutiS^" '"' ^^^^^««* - «-^004, ^a -0208 is the lu«t r 987G03 1975 all tlie quotients are • '-^ » 07 O-Ol I 0-0004 V ' The true quotient is 9890427lO4 », ^''•'^'** And the true ren.uindor 0.0908 ' Z u 7" "'' '^' .l"""""''- Uj 1 ^ u i'-u«, or tlie iast reniaindew or the part qf it ju,t UiSd lin \ ^»-«'n t^'e dividcul, tlio third lino 4701 ,nV J^''\ ^I"otient. Thus in as quotientrand ^Tof ^itf s'jl'rt' ^ ]T^ ^ ^^^^^ 701 us ren/ainde/ 4-7 wo ] ^'^^'^^^-that is, to the Icl/of tie decinn \^/' ;>^^^P.Vin^ four places, al pv- units as quj^;^ r^'al^^tlf^^ ^ ^^««' liue), one is a decimal i.!.. *! -^ ^ ^^" *^^ »ext tenths; and in OlO-JO InJt y \ ■ ^' "' ""= '"''ier four piaoos are doo m- iT f j '^'r'""" '™ "»' "^ 'he dreJths, &o. ""-'"»■"''. "« quotient must be huu- plied, and\he .:,,'';:;".";' '?"»/'- V-tient multi^ o-ddcd, be] onix o •oiiiaindcr to wliich the product t.ts IS to be 98 DIVISION. ^ 47 4H, 4!), 60, CI. 62. ca. ct, Go. (SQ. C7. 68. 61). 00. v(il. (52. (53. (54. G5. m. G7. 08. EXKHCtNKa. , r)0789-f.74l==70*7;« 47H!M>7-M)71=4l>;;JL'f. 1)77070 +'17(50(3=20 ?hZI. 607807 -^8 12=074 •"!• 78(57074 -f-071 2=816 W. JK)7O7()O-f-457O()O=(J-7l03. 07051 68 -^7894=8r)7. tt7470-^.;{l)00=17-3. 0OO0O-f-47()0O=l -4490. 70707 -^ 40700=1 -8802. (51M692-T-704;^24=8. 9070744-^9] 0070=1 • 0.'J29. 740070000 -^741000=998 -7449. 94 10(507 1 1 1 -f-45078=200043 • 1 1 32. 45407(5000-^400100=1 1 34-9003. 737(547(57(57 H-a46(570=:2i::;39 -049. 47 ;6782975-^20• 175=1 -8177. 47 -(355-^4 -5=10 -59. 760-98-^7G-7301 2=9 • 800. 75 -3470+8829=190 -7798. 0'l+7-0345=0-0000131. 5878+0-00090=5002083-33, &a 6J. If £7i)00 were to be divided between 5 persons. how nmch ought each person to receive > A71S. it; 1 500. 70. Divide 7560 acres of land between 15 persons. _,,,.., ^ , ^^?^. Each will have 504 acres. 71. Dmde £2880 between 60 persona. „r. rrr^ . , ^«-'- ^acU will reccivG £48. 72. >Vhat 13 the ninth of £972 ? Am. £108. 73. AVhat is each man's part if £972 be divided among 108 men ? Ans. £9. 74. Divide a legacy of £8520 between 294 persons. _. -^. „ -^ns. Each will have £29. 7o. Divide 340480 ounces of bread between 1792 persons Ans. Each person's share will be 190 ounces. /6. Ihere are said to be seven bells at Pekin, each ot which weighs 120,000 pounds ; if thev were melted up, how many such as great Tom of Lincoln, wcighinff 9894 pounds, or as the great bell of St. Paul's, in London, weighing 8400 pounds, could be made from them ? Ans. 84 like great Torn of Lincoln, with 8904 pounds left ; and lOOlilco the great bell of St. Paul's. 77, Mexico produced from the year 1790 to 1830 a I>IVIS10N. 09 ive £48. divided persons. Lvc £29. m 1792 ounces. :in, eacli ! melted vcighiug lul's, in ie from th 8904 Paul's. 1830 a f I" 5«7,01y,7-IO iuil,.3 i, ,' ",t ',- ' ""P"™ "l"'"' ifl29'0775 in 18'57 28b 13o6 m 1740; and I.alf of ,„en ?i I^ow , ?'" '"'• T ^ '"''"'"■'^ ■""• » * eo^:"ro:'L^;,t!:rf„,S:!i ^^ ^-^-'^ '^ I'-XAJii.LK.-Divldo 84380848 by 87532 87532)84380848(964 5G0204 350128 ' the dividend to vnlaindor after s,h..u.ttitl,o""'''.'l •*''^, *'^^^ ^«) ' 2 (tiio '"^ f'.'H-ried from 98 an.] 1 fT^^" ^? *!,'''°'"' '^ + '^''e 2 to rowed ulu.n „/ ' ^1 \ *" «<»"ir)en,siito for wl^.A ,„„ i.... lowta When wo coiwideved in ti e duidund as 10) ; (ti: 19 100 DIVISION. romalnder when we subtract tlie risht liand digit of 48 from 11 om the 48 )j (tho' remaindcu- after eubtractiii;]; the rio-hf hand digit ot 67 from 3, or rather 13 iu the dividend), "id {J times a + the G to bo carried from -the 07 + thel for what wo borrowed to make 3 in the dividend become 13) : dendT ""^"^^ '''' ^^^^'^ subtracting 79 fro^n 84 in the divi' j.n;t!,nf'f .^f *' '° .*¥ ^^^^".theses are merely explanatory, and not to be repeated, the whole process would be Jmstpart, 4, 18 ; o. 28 ; 2. 48 ; 07 ; G. 79 • 5 Second part, 8. 12; 2. 19 ; 1 32; 0. 45:5 53-3 lhirdparfc,8;0. 12; 0. 21 ; 0. iO;0. ^5 ; ' J he remainders m this case boincr cyphers, are omitted. All this will be very easy to the pupil who has prac tiscd what has been recommended [13, 23, and 651. I no chief exercise of the memory will consist in recol- lecting to add to the products of the diifcrent parts of the divisor by the digit of the quotient under consideration, what IS to be carried from the preceding product, and I mty bes.des-when the preceding digit of the dividend hnnd d?>"''r!u-'^ ^^ ^2 5 then to subtract the right hand di^it of this sum from the proper digit of the dividend (increased by 10 if necessary) QUESTIONS FOR THE PUPIL. 1 . What is division ? [66] 3. What is the sign of division ? [68] 4 How are quantities under the vinculum, or united ^^ TAr'°° ""^ multiplication, divided ? [69] 12 nn, r* r *V '?^? ""^'^ *^" ^^'"'' do«« not exceed 12 nor the dividend 12 times the divisor.? [70] piit Xn .f ' r'^ ^^^. ^^'' ''''''''' ^f '^' different paits, when the divisor does not exceed 12, but the divi.lend IS more than 12 times the divisor ? [72 and 771 7. How is division proved ? [79 and 95]. -•' b V\ liat are the rules wlion the dividend, divisor, or both contain cyphers or decimals.? [80]. 9. AV^hat is tho rulo •ii)<l wl--^ ^i--. -^k-. c •. ,.^ ";- ^""-7 •">5 wJiac are the reasons of 13 different parts, when the divisor exceeds 12 .? [89 and 93j . GREATEST COMMON MEASURE. 101 10. What is to be done with the remainder > r72 and 89j. ' '-''* 11. How is division proved by casting out the nines ? [95]. 12. How may division be abbreviated, when there are ^lecimals i' [96] . ' 33. How is division performed, when the divisor is a composite number ? [97] . • v^; v!T ,'^ *^^ division performed, when the divisor i.s but little less than a number which may be expressed by unity and cyphers ? [98] . 15. Exemplify a very brief mode of performin<r divi- sion. [99]. / ° THE GREATEST COMMOxY MEASURE OF NUMBERS 100.^ To find the greatest common measure of two quantities — IluLE.— Divide the larger by the smaller ; then the divisor by the remainder ; next the preceding divisor by the new remainder :-continuo this process until nothing remams, and the last divisor will be the greatest common measure. If this be unity, the ffiven numbers arQ prime to eack other. and 4248''' ~^'"*^ *^® greatest common measure of 3252 '"va ?)4248(1 3252 99G)3252('3 2988 264)990(3 792 204)264(1 204 60)204(3 180 "24)6012 48 12)24(2 24 V im 102 GREATEST COMMOxV MEASURE. fh^i'.*^"'^^''^* remainder, becomes the second divisor 2G4 the second remamder, becomes the third diW&o 19 the last d.v.sor, is the required greatest cLnrS^easuro ^ tied that ' f any quantUv m^ZZ^ ^'' A^' ^^^^^ *° ^^^ «a««- any multiple of t^,2t Xr » thus ff T^n'-': %f^} ^^^^'^^^ sum. for if 6 g. into 24, 4 timet and^iS^ ? tL^U ;n/evt dently go into 24+36. 4+6 times :--that is, if ^=^, ^nd^^- 6,-6-+?=4+6. • 6 6 ence between the numbers of ti,^P«u V ^*^^ ""^ *^« ^i^er- due to this difference, l^stte^TtalLr!^ ft^^e ^^ of tmies ..-that is, 8ince^i=6, and ^=4, ?^_24 X 36441 oth^rliSSCr ^'^^^^^ e^TeSIHorrect with any and that it is L ^2^ folo'n 1^^ ^ "^^"^^ ^ we fiXraH 2leTsrer24'^tf"^ ^* *^^ ^"^ «^ t^^« Process, a multiple of 24?^ thl'slTiSw?' ^'' ^f^^^^ ^* '^ each of them) or 60- and IftThlv 4^-^''^''^® '* measures and 180+24 Twe hnv^ S^ • . because it is a multiple of 60 • these) or ib4 an Jo^teo-'or 2rT^ that it measures each of pie of 264; and ?92+2ot or 996 ^'hS^QkI^' ^'''''.'' ^ ^^^l^" and 2988+264 or 3252 Innp J vl' ■ ^^^^' * multiple of 996 : 996 or 4218 (throJlSi v'erLmL?)''^^.^,^^^)!"^ '''^+ each of the given numbl-s an<f^ fS ^^^^^^'^^^^ it measures some other be greater ir/n ?tl ^'.^'"^on measure. If not, let process) measu^r7nf4248 and s'S'TtTS 'T."' *^^ ''^ ^' '^' measures their difference 996 aniio««' k^' supposition), it of 996; and. because irmefur'es 3252 n'^^S*^"^ "^^^*^Pl<^ their difference, 284 • an.l 7Q9 tL ' '^'''^ ,^^^^» ^* measures the difference between j'6 and Tgr^'^of^^^^^^^^^ ^'^^ between 26 4 and 204 or GO • and 1 «0 h ' ""'^ *''^ difference . ami the difference between''-'04 and 1Ro"''"h *" '"^^*'P^« ^^ ««; a multiple of 24; and tre'd^ffl^l ^'^t^^A^ '^^^^^^^ But measunng 12. it cannot be greatc^,- tha^ 12 """ "° "^ ^' GREATIuST COJIMOxV MEASURE. 103 rntJre\TZlZnZZi'f'T.' *^* '^"^ other common sequently that 12 s tH 'IS' n ^''' *^''^ ^^~'"^^ ''«"- rule might be in-ovea fiom r!nv !> '"°" measure. As the it is true in all Lses ^ °*^'''' ^'^^"^JP^^ ^^"^"y well. ^ 104. We may here remark, that the measure of two , -my quantity, tiio diitit of whoso bwest donnmin„n„„ .3 au even .mn.bor i« aivimhh by 2 at ka t '"" Any number on, ,ng in 5 is divisiblo by 5 at least j^ Any nttmber onding i„ a cypher is div^isib?^ l'; io at EXERCISES. ancl-lSSV"'lt.f ''' ''"^"^"'^ "^^^'^"^^ •^f 464320 2. Of 638296 and 33SS8 ? ^,w q 3. Of 18996 and 29932? ^t 4 4. Of 2G0424 and 54423 > Ans 9 5. Of 143168 and 2064888 .P Ans'. 8. t). Of 1141874 and 19823208 > A^Lf. 2. ihlni^'Jl'Jl ^'''''''' — «« --sure of more of lh4 1 If ''^'"''"''^ "^"'^•''^^•^ a»d a third ; next ot this hist common measure and a fourth &e S last common measure found will v.. +i "'^"' ^'^^ J-^ie measure of all the given ^il^^^^ '^' ^''''''' ''^^'^ SOotTnd 673^^^'''^ '^'' ^''^'''^ «^°^^«° «^oasure of 679, .n-eatost common measure of 7and 6731 71?' *''^' '^' ■'''" numhor), for 6734 -^7-060 wIH.^? • ^^'"^ remaining 7 is the required numbTr ' ""^ ^•'^•"'•^•igli'. ThereforS TSoi'^mfHil '"^ "'^ '''' -^^^^^^t ««^'"non ;^a«ure of 0:^0, 104 LEAST C0iv/1\I0N JMULTIPLE. The greatest common measure of 93G and 73G is 8 and tlic eonnnon measure of 8 and 142 is 2; therefore 2 is 2o groutesfc common measure of the given numberr fncf n toCfa'ft'^ .go through all of themin succession ; nro to l..S,r„T*i * '* '^ ^^^ greatest common measure, we used toind tU .T'""'"''''^'"""* '^ ^^^ fi^'«t process, or tlm? usta to hnd the common measure of the two first numbers tond proceed successively through all. numDers, EXERCISES. i'tLo^^^^jH^^ greatest common measure of 29472 176832, and 1074. Arts. 6. -»^'^, 8. Of 648485, 10810, 3672835, and 473580. Am 5 9. Of 16264, 14816, 8600, 75288, and 8472. Ts 8. THE LEAST COMMON MULTIPLE CF NUMBERS. titils— ^"^ ^""^ ^^"^ ^'''''* '°'''"'°'' """"^^^P^^ °^ ^^'^ ^^^"- lluLE.— Divide their product by their greatest com- mon measure. Or ; divide one of them by their greate t common measure, and multiply the quotient by the o lier-the result of either method will be the required least common multiple. " 4""^"- KvAMPLE.-Find the least common multiple of 72 and 84, 1-. is^their greatest common measure. 1^ = 6, and G X 84 = 504, the number sought. 108 Reason of the Rui.E.-It is evident that if we muU SiniT'^i "^""^'r^ *°Sether, their product wHl be a tiultiple of each by the other [30]. It will bo easv to find he smallest part of this product, which will stiKe their of^r'nniilMB- '-'71 r^"""'^^' '''" 1^^'^^' ^'^^^^ of t'^e factors Irotlot^nf^Xr f ^ ''"•X ""'"^^^' '^"'i multiplied by the & J -V 1 ,""'■, ^'"'^'''■'' '' "^'""^ to the product of all the .io-.« ^n-idcl by the same number, ironco "'>. .pd ri L; ! LEAST COMMON MULTIPLE. 106 $m 2X84 ~^-^ (the nineteenth part of their producc)=I?x84, or 72 x J. y u _. Now if '-- and __ be equivalent to integers, ^x84 will be a multiple of 84, and°|x72, will be a multiple of 72 [29] ; andi--g_, L.X84, and 72 X~| will each be the common multiple of 72 and 84 [30]. But unless 19 is a common measure of 72 and 84, j-^ and _ cannot be both equivalent to integers. Tl.erefoie the quantity by which we divide the product of the gixen numbers, or one of them, before we multiply it by the ctlior to obtain a new, and less multiple of them must be he c.mmon measure of both. And tbe multiple we obtain w 11 cvi.lent y, be the least, when the diviso ■ we select ^stlo grea es quantity we can use for tiie purpose-that is! e greatest common measure of the given numbers It follows, that the least common multiple of two numbers, prmie to each other, is their product. EXERCISES, 1. Find the least common multiple of 7S and 93 Ans. 2418. ^ 2. Of 19 and 72. Ana. 1368. 3. Of 464320 and 18945. Ans. 1759308480 4. Of 638296 and 33888. Ans. 2703821856." 5. Of 18996 and 29932. Ans. 142147068. 6. Of 260424 and 54423. yl%5. 1574783928. 109. To find the least common multiple of three or more numbers — KuLE.— Find the least common multiple of two of thcin ; then of this common multiple, and a third ; next ot this last common multiple and a fourth, &o The last common multiple found, will be the least common multiple sought. TA-AMPLE.-Find the least common multiple of 9, 3, and 27 ^ '^ IS the greatest common measure of 9 and 3 ; therefore g X 3, or 9 is the least common multiple of 9 and 3. '^^9 is the greatest common measure of 9 a«7 ; therefore ^ X 9, or 27 is the required least common multiple. I, ii J; iii 106 LEAST COMMON MULTIPLE. that^^ili'lir/"'.'""*^ RuLE.-By the last rule it is evident tnat J7 IS tl e least common multiple of 9 and 27. But since a nuauri* %"•' 'IV'V, ^^'"/'^ -.^^ multiple of U, mu^t^airt: tlii is Imanif ."'"'i" ^'"''' ""'""^"^ "^"^*'P^«' ^e^a'^so none tnat IS smaller can be common, also, to both 9 and 27 since they were found to have 27 us their least common multiple EXERCISES. it aIs. mls'^'^ ^'''''^ ''''"'""'''' multiple of 18, 17, and 43. n Ri .^?' 3 ^^^ ^"^ ^1- '^>^'- 1265628. 10 ^;^^-«?f ?'.rp,f ^' ^^^ ^^^2. .in.. 2937002688. 10. Of 53/842, 1G81<J, 4367, and 2473. 11 Of oir'jp o.iot. n... ^«5. 8881156168989038. li- Of 21636, 241816, 8669, 97528, and 1847 Ans. 1528835550537452616. ' QUESTIONS 1. IIow is tho greatest common measure of two quan- tities found .? [louj. ^ 2. What pvinciples are necessary to prove the correct- ness ot the rule ; and how is it proved ? [101, &c 1 o. llow IS the greatest common measui-e of thi-ee, or more quantities found .? [105]. ' 4. How is the rule proved to be correct ? [1061 o liow do we find the least common multiple of two numbois that are composite .? [107]. 6. Prove the rule to be correct [lOS]. 7. How do we find the least common multiple of two prime numbers .? [108.] 5. How is tho least common multiple of three or more numbers found.? [109]. 9. Prove the ;ule to be correct [110]. ^ In future it will be taken for granted that the puB^ IS to be asked the reasons for each rule, &c. t 107 SECTION III. 19 1 m quan- REDUCTION AND THE COMPOUND RULES. The pupil should now be made familiar with most of the tables given at the commencement of this treatise. "^ -REDUCTION. t 1. Reduction enables us to change quantities from one '^nomination to another without altering their value. Taken in its more extended sense, we have often nractised it already : — thus we have changed units into tens, and tens into units, &c. ; but, considered as u separate rule, it is restricted to applicate numbers, and is not C0nfin|4 to a change from one denomination to the Tiexi higjier, or lower 2. Reduction i» either descending, or ascending. It is reduction descending when the quantities are changed from a higher to a lower denomination ; and reduction ascending when from a lower to a higher. JRjcduction Descending, 3. RuLE.^^Multiply the highest given denomination by that quantity which expresses the number of the next lower contained in one of its units ; and add to the product that" number of the next lower denomina- tion which is found in the quantity to be reduced. Proceed in the same way with the result ; and continue tlie process until the required denomination is obtained. Example. — Reduce £6 16s. OjcZ. to farthings. £> s. d. 6 „ 16 „ Q\ 136 shilling8=£6 „ 16. 12 1632 pence = £6 „ 16 „ 0. 4 C520 farthings OB jC6 „ 16 „ Q\. " * 1 l a— l: :^'i " 11 ■! 108 REDUCTION. 6 aro 24, and 1 nrA 9*^ 4 .. ^"^ ^ ^^°?6« ^ ^ o 12. 4 tunes 4 SrS m'ny' fertile ."^"^"^'""'^.■'™™• pence to EXERCISES. 93312^'''" '^''''^ ^^'^^""°' ^*" ^^^~S P^^^e-^ ^^«. 2. How many shillings in i2348 > Ans. 6960 ^. How many pence in ^638 10^. ? Ans. 9240 4. How many pence in ^58 13.. ? Ans. 14076 5. How many farthings in £58 135. ? Ans. 56304 67291 "'"""^ farthings in £59 13.. G^d. ? Am, 7. How many pence in £63 0.. 9d. ? Ans. 15129. a!s\ 1864. ""'"^ P'""^^ ^" '' ^^*-> ^ ^^^-^ 16 ife- -^ .l.^: 1^68. '"''''^ ^'"''^' ^"^ ^'^ '^*-' ^ "i"'- 1^ J'^- -^ ^^^5^, ?°^ "^^"y grains m 3 lb., 5 oz., 12 dwt- 16 IlKDUCTION. 109 11. How many grains in 7 lb., U oz., irrdwfc./M grains ? Aiis. 45974. 12. How many hours in 20 (common) years? Am 175200. 13. How many feet in 1 English mile ? Ans. 5280 14. How many feet in 1 Irish mile > A7is. 6720 15. How many gallons in 65 tuns .? Ans. 16380 16. How many minutes in 46 years, 21 days, 8 hours, 56 mmutos (not takmg leap years into account) ? Ans. *4208376. 17. How many square yards in 74 square English perches ? Ans. 2238-5 (2238 and one half). IS. How many square inches in 97 square L'ish perch- es.? Ans. 6} 59H88. ^ 19. How many square yards in 46 English acres, 3 roods, 12 perches ? Ans. 226633. 20. How many square acres in 767 square English miles ? Ans. 490880. ^ 21 . How many cubic inches in 767 cubic feet ? Ans. I32o376. 22. How many quarts in 767 pecks .? Ans. 6136 23. How many pottles in 797 pecks ? A71S. 3188. Reduction Ascendviw. 5. required denomination is . hings to pounda, &c. EuLE.— Divide the given quantity by that number ot its units which is required to make one of the next Higher denomination— the remainder, if any, will be of the denomination to bo reduced. Proceed in tlie same manner until thi ' ' 1 o ■ . , , obtained. Example.— Reduce 8(.,, 4)856347 12)214086f- 892 ; „ 01=856347 farthings. 4 divided into 85G347 farthings, gives 214086 ponce arul 3 farthings 12 divided into 2!4ol6 pence, gives 17840 'i«.lo°^' r^ ^ ir^V,9«- 20 divided into 1^840 shillings, a<ives 7}7xr ''" i""" «^iiJ^ings; there ie, therefore, nothing in the Bhilhngs' plac . of the result. ** no HKDUCTION. Wy (livi.|r, by 20 if ^0 (llvi.lo l.y 10 lui.l 2 FSoc If 971 any, [N,o. 34] w uch will then bo tbe unit« of sbilli, .^s 11 t b., result ; and tie quotient will bo tens of shillings :^ i.viding tbe xa tor by 2 gives the pounds as quotion? and di;;:srlS^'^' '' ''^^^ ^-^ -^ - ^^- Quired q.«.. VoIIk-^"''"'"'" "*^ T"^ Rin.K.-It is evident that every 4 Earthings are equivalent to one penny, and every 12 pence to «neslnll.ng,&c ; and that ^hat is left after taking away 4 far b.nga as often as possible from the farthings? intTsfbu farthings, what remains after taking away 12 pence as often as possible from the pence, must be pence, &o ^ 7. To prove i?€r/%c^207i.— Reduction asccndin'' and acscendiDg prove each otljer. ^ Reduction < f * s. 20 417 12 d. farthirgs. 4)20025 Reduction j 12)5006| 20)417'„ 2 5006 4 4)20025 Proof . Proofs 1 12 )5006| 2 0)417 „ 2 ^20 „ 17 „ 2] ^20 „ 17„ 2« 20 417 12 20025 farthings. RXERCISES. nooto ^^"""^ "'''''^ P^'^^^ ^" ^3312 farthings ? Ans. 25. How many pounds in 6960 shillings? Ans ^2348 roon a7 ""^""^ P''''''^'' ^'^^ ^^ ^^^ halfpence ? Ans. pi/fi Us, od. JI\ ^""Z^^"^^ ^^^"^^^^ &c. in 7675 halfpence ? Ans. Jblo 19*. 9^a. 28. How many ounces, and nounds in 4352 drams? Alls, 2/2 02., or iv ib. REDUCTION. Ill 29 How many cwt., qis., and pouuda in 1864 pounds > Ans. l(i cwt.,2(ir,s., 1(5 lb. i -". 30. How many hundreds, &c., iu 16G8 pound.s. A,ls 14 cwt., 3 qns., i6 lb. ^ 31. How many pounds Troy iu 115200 frvahiH > Ans. 20. " b • J ^^vIm'^^ '""""^ 1'°''°'^'* '" ^^'^''^20 ^^' avoirdupoise ? tills. d720. .,.2^1* ^?^^ J"*"^ lio<rsheads in 20C58 gallons ? Ans. 127 hogsheads, 57 gallons. 34. How many days in 87G0 hours ? Ans. 365 35. How many Irish miles iu 1S34560 feet.? Ans. i lO. ^^- ^low many English miles in 17297280 inches ? 37 How many English miles, &c. in 4147 yards > Ans. 2 miles, 2 furlongs, 34 perches. iiS. How many Irish miles, &c. in 4247 yards ? Ans i mile, 7 furlongs, 6 perches, 5 yards. 39. How many English ells iu 576 nails ? Ans 28 «ils, 4 qrs. 40. How many English acres, &c. in 6097 square yards ^ A71S. 1 acre, 8 perches, 15 yards. 41 How many Irish acres, &c. in 5097 square yards ? Ans. 2 roods, 24 perches, 1 yard. . 42. How many cubic feet, &c., in 1674674 cubio inches ? Ans. 969 feet, 242 inches. 43. How many yards iu 767 Flemish ells ? Ans 07o yards, 1 quarter. 44. How many French ells in 576 English ? Ans. 480 .f i/\ T^J^^ J'^'-^^-' *^^" '''^^ ^^'"^^^^ of a pound of gold, to farthmgs ? Ans. 44856 farthin^^s 46 The force of a man has been estimated as equal to what, in turning a winch, would raise 256 lb in ?'"''?nl'/^'^ ^^' ^" ^'"^Sing a bell, 572 lb, and in row- mg, 608 lb, 3281 foet in a day. Uow man>^ hundreds quarters, &c., in the sum of all these quantities ? An^' 16 cwt., 2 qrs., 7 lb. 47. How many linos in the sum of 900 foet, tha i 112 nzDvvvioti. length of ♦ho tomple of tho sun at Balboc, 450 foot its breadth, 22 foot tho circuruforuiiee, and 72 feet tho height of many of it« columns ? ylns. 207936 , 48. How many square toot in 7tJ0 English acres, tho inclosuro m which tho porcelain pagoda, at Nan-Kiiiir, 4J. rho great boll of Moscow, now lying in a rit qfionnnT /"'•' supported it having boon burned, weighs 36)000 lb. (some say much more) ; how many tons, &o , m this quantity ? A7i^. 160 tons, 14 cwt., 1 qr., 4 lb. QUESTIONS FOR THE PUPIL. 1. What is reduction .' [1]. ^ 2. What is the difference between reduction descend- ing and reduction ascending > [2] . 3. What is tho rule for induction descending ? fal 4. What IS tho rule for reduction ascending ? fsl 5. How is reduction proved ? [7]. Qiiestiom fonmdcd on the Talk page 3, c^. 6. How are pounds reduced to farthings, and farthincrs to pounds, &c. ? ^ ' t3 7. How are tons reduced to drams, and drams tc tons, &c. .'' 8. How arc. Troy pounds reduced to grains, and grams to Troy pounds, &c. } b j ^ 9 How aro pounds reduced to grains (apothecaries weight), and grams to pounds, &c. ? 10 How are Flemish, English, or French ells, re- eHs'' &o'' T ' ""'* '"''''''''' ^'^ i^lemish, English, or French ^^11. How are yards reduced to ells, or ells to yards, 12. How arc Irish or English miles reduced to linos, or lines to Irish or English miles, &c. > ' 13. How are Irish or English square miles reduced to square mehes, or square inches to Irish or English square miles, &c. ? ^ n 450 foot its 72 feot tho m. 1 acres, tho Nan-King, 35000. [? in a j)It nod, weighs y tons, &o., qr., 4 lb. COMPOUND RULES. 113 a descend- er ? [5]. S^'C. 1 farthings drains tc lins, and itliccarios ells, rc- >r French to yards, to linos, reduced English 14. Ilowaro cubic feet reduced to cubic inches or cu»)io inches to cubio feet, &c ^ "loncs, or U,,!i;-«.'':T "'•" """' ""'"™^' "• """«-■'. or naggin, .o lu,',»; "■"' "™ •'""'' "^""«^ '" 8"''™^ <" gal'ons to and';:,":: i:JX ^''^ •"■•"--) -J"-'' to pint,, yoa'^i &o!" ""■" ^"™ '"''"""'' '" ""^•'»- 0'- tlmU, to or Jw'"t:i';;;!:f:T<,.<f "■» <='™'") -^'-o'' '° tWrd., THE COMPOUND RULES. ^. The Oonipound Rules, are those which relate to apphcate nuinbcvs of more than one denomination.' *' conin 1 i " ^^ '"o.u^y, wcnghts, and measures, wore coast, ucted according to tlie decimal sv.stcm, on v the T ?' ^^S^"P^« ^^^^'^'tion, &c., would be' Sir d to d Z \% ' ««"«i^l«'-ble advantage, and ^ a '' tond to snuplify mercantile transactions -If i o f • • things were one penny, lu pence one shillincr and To shil in,g. one pcnind the addition, for exan";!^ f i? a noun i \l \^^'- u''^- ^V'''' ^^^^^^^^ to eparate Id OikU r ^'" "?^ of comparison," from its parts! bots'follow^^^^^-"^ * " ' *^^^^^^ '' ^ P-"^)> '-Id 1-983 6-865 Sum, 8-848 The addition might be performed by tho ordinarv rules, and the sum read off as follows " mV . ^'^'^'"''/y ci^hfc sliillings, four pence, ilnd ^ 7.,^ ^^ K^'j; ^ even with the present arrangemoiU of mo ; wei. i tt^ir^ the rules alr^uly given for ad^Ji""^^' ttr un-/^ ' ""'f"^ "'"^^^ liave.b.en made to include he a d.tion subtraction, &c., of .r.nlicate .^ZZ can...tmg of more tlian one denomination ; sin"cc"'thQ l! ¥■ p w 114 COMPOUND ADDITION. principles of both simple and coiDpound rules are pre- cisely the sa.no-the only thing necessary T bcnr carefully m n.md, being the number of any mo de! nomination necessary to constitute a unit o/tl next COMPOUND ADDITIOX. i^nl' nfl'^^'~^' ^i''^ ^'''7'' ^^^« ^'^^^euds so that qnantU vertical oIuZ ''T'T'''' "^'^^^ «^^^"'^ ^'^ ^^« --« wi nf ^^^•^"^n-units of pence, for instance, under units of pence tens of pence under tens of pence un ts 01 shilWs under units of shillings, &c ^ ' U. Draw a separating line under the addends. aenoLi^it:^,:^^!^^^^ the same pence, ^., begSnin^wilh IhrLtst^^^^'""^^' ^^"^^ '^ ber of tl .rl? '^"\^^^;;«J c«'""m be less tlian tlie num- toei ot that denomination which makes one of the m vf ^ s. d. 52 17 33 ) 6} } addends. 47 60 5 14 2' \ IGG 17 ! and I make 3 farthings, which w^Mi ^ m..i.« r r -a 2 are 3, and « arc 9, aad 3 are 12"pro/i:;,'uaf tJ'S Ill OS are pre- !<'iry to bear any one do- of the next that qnanti- iu tho same mice, imJer pouco, units mds. f the same ;«, pouco to n tlic nura- )f tho next ot, for each nomination one to tho under the nomination it in the 47 5s. G}/1., ike () far- denoiuiiiii- 10 jiresont. uarricii) ual to one COMPOUND ADDITION. Hq of the next denomination or thof nf c1,-it . -, and no pence to be sriow 1 t^r""'' ^''^ ^^"'i'^d, in the pence' place of the su n 1 sti r ""''z?",^ '*' ^^^P^*^^ and 14 are 15,^and 5 are 20 ^d 17 n i -^ i-u-^^ ''^'''^^) to one of tl. next ck^nomLtl or hat'of n "?.'~r^r earned, and 17 of the present or thnfViFn""'*'' *^ ^^ set down. 1 pound anf 6 l^eS^'andl are if 3' 9^ ^' 10 pounds— equal to unit, nf Z^ j ^ . • "'"' 2 are 1 t'u of pounds be cm ed l"^?™ ' w''" "'',''''''"■ »d 11 and 5 L 16 tens ofTonSd^ VZ^tZT ' ""' " "" thoVe-aS;t;or[st r;Xt It' TT not so necpssflrv fn %.„+ T -i' ■•■* ^s evidently Of all the sunis. "^ atterwards find tne amount Example : £ s. d. 57 14 21 32 10 4 19 17 6 8 14 2 32 5 9j 47 6 4) 32 17 2 5(J 3 9 27 4 2r 52 4 4 37 8 2 = 151 7 11 ^ s. d. 404 11 10. = 253 3 11 a dlt .?'nfr '''^'^'''^ "'"'^ ^°^"^"' ^« "laj put down «rpl?-l' ""'""" ""^ '- o™- dot- "U^ing ti; S I I 116 COMPOUND RULES. & s. (/. 67 •14 2 32 10 4 19 •17 •G 8 •14 2 32 5 •9 47 •6 4 32 17 2 56 •3 •9 27 4 2 52 4 4 37 8 2 404 11 10 2 pence and 4 are 6, and 2 are 8, and 9 are 17 pence- equal to 1 shilling and 5 pence ; we put down a dot and carry ?• , ^,«:"4 2 are 7, and 4 are 11, and 9 are 20 pence-equal to 1 shilling and 8 pence; we put down a dot and carry 8. « and 2 are 10 and are 16 pence-equal to 1 shillino; and 4 pence ; we put down a dot and carry 4. 4 and 4 are 8 and 'T"" 1^— which, being less than 1 shilling, wo set down under the column of pence, to which it belongs, &c. We find on addmg them up, that there are three dots: Ave therefore carry o to the column of shillings. 3 shillings and 8 are 11, and 4 are If, and 4 are 19, and 3 are 22 shillings-equal to 1 pound and 2 shilhngs; we put down a dot and carry 1. 1 and 1< are 18, &c. ^ Care is necessary, lest the dots, not being distinctly marked, may be considered as either too few, or too many. Thia method, though now but little used, seems a convenient one. 14. Or, lastly, set down the sums of the farthinfrs, shillmgs,_&c., under their respective columns; divide the tarthmgs by 4, put the quotient under the sum of tlio pence, and the remainder, if any, in a place set apart tor It m the sum— under the columu of farthings : add together the quotient obtained from the farthiSgs and the sum of the pence, and placing the amount under the pence, divide it by 12 ; put the quotient under the sum of the shilbngs, and the remainder, if any, in a place allotted to it in the sura— under the column of pence ; add the last quotient and the sum of the shil- lings, and putting under them their sum, divide tho latter by 20, set down tho quotient under the sum of COMPOUND ADDITION. 117 17 pence— •t and ciu-ry nee — equa! id carry 8. hilling and I are 8 and ) set down . We find, e therefore d 8 are 11, — equal to id carry 1. ly marked, -ny. This snient one. farthinfi:s, i ; divide um of tlio set apart ngs; add lings and nt under mder the my, in a )lumn of the shil- vide tho sum of ■ the pounds, and put tho remainder, if any, in the sum— under the column of shillings ; add the last quotient and tho sum of the pounds, and put the result under the pounds. Using the following example — £> s. d. 47 9 21 362 4 in 51 16 2| 97 4 G 541 13 2i 475 6 4 6 11 11.1 72 19 9,^ 1G51 82 47 13 farthings. 4 4 3 86 50 1055 G 2!- The sum of the forthingg is 13, which, divided by 4, give.i 3 its quotient (to be put dowia under the pence), and ouh farthing as remainder (to be put in the sum total— under the farthings). 3,'/. (the quotient from the forthings) and 47 (the smn of tlie ponoc) are 50 pence, which, being put down and divided by 12, gives 4 shillings (to be set down under the shillings), and 2 pence (to be set down in tliu sum total— under the ponce). 4.^. (the quotient from the ponce) and 82 (the sum of the shillings) are 86 shillings, which, being sot down and divided by 20, gives 4 pounds (to be set-^down inider the pounds), and G shillings (to bo sot down in the sum total— under tho shillings). £4 (tho quotient from tho shillings) and IfuU (the sum of tho pounds) aro 1G55 pounds (to be set down iu the sum total-- under the pounds). The sum of the advlendss ?s, therefore, found to be j£lG55 6s. 2|(/. 15. In proving the compound rulo«(, wo can geuorally avail ourselves of tho methods used with the sin.i^l<? vul,^ [Sec. IT. 10, &c.] 11 m ssM (18 COMPOUND ADDITJON. £ 8. d. 70 4 6 57 9 9 49 10 8 183 4 11 £ s. d 674 14 7 466 17 8 676 19 8 627 4 2 KXERCISES FOR THE PUPII, Money. d. £ s. d. 7 76 14 7 6 67 16 9 8 76 19 10 £ 58 14 69 16 72 14 * s. d. 767 15 472 14 567 16 6 6 7 423 3 10 £ s. 567 14 476 16 647 17 527 14 d. 7 6 6 3 (4) £ ». d. 84 8. 2 96 4 Oi 41 6 (8) £ .V. d. 327 8 6 601 2 111 864 6 121 9 84 £ s. d. 4567 14 6 776 16 7 76 17 9 51 10 44 5 6 (10) £ s. a 76 14 7 667 13 6 67 16 7 6 4 5 3 2 4 ^ s. d. 3767 13 11 4678 14 10 767 12 10 11 8 4 9 6 11 (12) £ s. 6674 17 4767 3466 6r^4 8762 d 16 Hi 17 101 2 24 9 9 £ s. d. £ s d 9767 6i 6767 11 ei 7649 11 2i 7676 16 94 4767 16 101 5948 17 sl 164 1 1 6786 7 6 92 7 24 6326 8 24 (15) £ s. d. 6764 17 6| 7457 16 5 6743 18 04 67 6 6k 432 6 9 « (16) £ s. d, 634 7 114 65 7 7 7 12 lOi 5678 18 8 439 « (17) * s. d. 14 71 677 1 6767 2 6 8697 14 74 6684 0| (18) * s. d. 5674 16 7i 4767 17 61 1645 19 7i 3246 17 6 4766 10 5| (19) £ 8. d. 6674 1 94 4767 11 10^ 78 18 Hi 19 104 6044 4 1 £ 4767 14 743 13 7674 14 7 13 760 6 (20) *. d. 7i 74 6i 84 4 5( 6i 34 COMPOUND ADDITION, 119 (21) £> s. d. 674 11 11.1 667 14 10| 476 4 11 347 15 Oi 476 18 94 (25) £ s. d. 576 4 7 7 732 19 04 667 9^ 764 2 64 n 6 (22) £> s. d 476 14 576 15 76 17 576 11 463 14 7 H n 8 94 (26) * s. d. 549 4 6i 7 19 91 16 64 734 19 9i 666 14 44 (23) £ s. d. 674 13 Bk 45 16 74 476 4 61 577 16 04 678 6 8| (27) £ s. d. 876 3 5 66 11 11 123 6 24 12 (241 £ s. d. 674 17 6^ 123 12 2 667 7* 679 18 91 476 6 64 (28) — •• £ 8. d 219" 6 32 11 8^ 04 127 8 2 29 6 6i (29) Jlvoirdupoiae Weight. (30) (31) "nT ^l T/-¥ ,5 ^-^-^^l- .^ cwt.qS «> 37 14 2 1 14 44 15 66 11 47 1 3 1 16 11 16 34 3 17 66 37 1 16 57 47 2 27 3 14 1 17 58 2 26 128 12 (33) (34) (35) (86) tT^^.t'^-iur.r^.^j.^-v-- 66 3 r i» 13 69 2 17 20 476 764 3 1 47 2 17 3 6 o 81 2 14 67 1 15 1 15 667 2 19 7 4 1 20 14 67 3 2 18 767 1 n 7i 74 6i 34 4 777'- T ,■? i?.'-F%i? .T'-^^ J> -.^S 767 44 567 676 341 1 1 3 1 2 16 476 "1 24i 447 J 7 766 3 214 676 13 767 1 16 467 667 2 15 563 11 973 1 12 428 1 1 1 1 7 6 7i 6 04 14 12 8 4 7 5 fl> 12 7 15 8 14 IS t,r 130 COMPOUND ADDITION. lb 7 6 9 ib 67 07 66 74 12 (41) Troy IVcight. (42) °n" *^7*- ^ff • ? °^- d^t- grs. lb U 6 9 6 9 - o- 6 6 6 7 8 8 7 7 6 G 7 4 88 80 (43; OB. dwt. gra. 7 9 ^8 9 8 8 7 6 21 11 18 9 9 8 6 3 (44) dwt 12 11 10 6 6 (45) 14 87 11 5 3 44 4 07 oz. dwt. gfs. lb \J 7 12 67 11 12 3 16 14 40 12 10 13 22 8 9 10 11 (40) oz. dwt. grs. 10 14 ^1 11 :. 9 7 e 18 14 9 8 10 (47) yds. qrs. nls. 99 3 1 47 1 3 70 3 2 Cloth Measure. (48) yds. qrs. nls. 176 3 3 47 2 7 3 3 (49) (50) y^'- *^o^- "S^' y^«- <!"• nls- 37 3 2 2 1 2 3 2 224 6 3 2 3 (63) i,51) (52) UTlTTSiiifr 54 3 673 2 3 ts. *»9 80 98 87 41 407 1 173 148 92 1 2 1 3 2 (55) hhds. gls. ts. 3 9 89 39 7 3 40 70 2 27 44 1 20 64 TVine Measure. (56) hhds. gls. 3 3 3 4 1 56 2 7 2 17 (57) ts. hhds. gls. 76 3 4 67 3 44 1 66 5 3 4 G02 27 sn CX)MPOUND ADDITION. 121 (43; E. dwt. grs. 7 9 8 9 8 i 7 6 (46) . dwt. grs. > 14 11 11 ^• 9 7 5 18 14 . (50) as. qrs. nls. 2 1 6 3 2 3 (54) i. qrs. nla. 6 1 1 6 3 1 110 3 2 3 57) ihds. gla. 4 44 66 4 27 8 8 1 3 Time. (58) (59) (60) ^99 st \'- Z ff • t K'- ^f: y- ± ^-. -. 99 859 9 88 8 77 120 7 66 67 49 265 115 2 42 60 6 90 76 1 3 1 2 60 67 68 69 127 7 120 9 70 121 11 6 47 3 8 9 11 60 44 44 41 17 61. What is the sum of the following :— three hun- dred and ninety-six pounds four shillings and two pence • five hundred and seventy-three pounds and four pence halfpenny ; twenty-two pounds and three halfpence • four thousand and five pounds six shillings and three farthings.? Ans. iE4996 IO5. S^d. 62. A owes to B ^£567 16s. 7Jrf. ; to ^£47 I65 • and to D ^56 Id. How much does he owe in all ? Ans. iE671 12s. S^d. 63. A man has owing to him the following sums • ^3 10s. 7d. ■ £46 l\d. ; and ^52 14s. U. How much IS the entire .? Am. £102 5s. ^\d. 64. A merchant sends ofi" the following quantities of hutter :— 47 cwt., 2 qrs., 7 ft, ; 38 cwt., 3 qrs., 8 lb ; and 16 cwt., 2 qrs., 20 lb. How much did he send off m all .? Ans. 103 cwt., 7lb. 65. A merchant receives the following quantities of tallow, viz., 13 cwt., 1 qr., 6 ib ; 10 cwt., 3 qrs., 10 ft,: and 9 cwt., 1 qr., 15 ft,. How much has he received in all.? ^?is. 33 cwt., 2 qrs., 3 ib. 66. A silversmith has 7 ft,, 8 oz., 16 dwt. ; 9 lb 7 oz., 3 dwt. ; and 4 ib, 1 dwt. What quantity has he > Ans. 21 ib, 4 oz. ^ j 67. A merchant sells to A 76 yards, 3 quarters, 2 nails ; to B 90 yards, 3 quarters, 3 nails ; and to C, 190 yards, 1 nail. How much has he sold in all .? Ans 357 yards, 3 quarters, 2 nails. 68. A wine merchant receives from his corj^spondent 4 tuns, 2 hogsheads ; 5 tuns, 3 hogsheads ; and 7 tuns, 1 hogshead. How much is the entire > Ans. 17 tuns 2 hogsheads. ' II; 122 COMPOUND ADDITION. .69. A man has three farms, the first contains 120 vf ?' on '''°^?' '^ P^'-^h^^; the second, 150 acres, 3 roods 20 perches ; and the third, 200 acres. How much land does he possess in aU ? Ans. 471 acres, 1 rood, 27 perches. ' ^.^u, ^/ 70. A servant has had three masters ; with the first He lived 2 years and 9 months; with the second, 7 years and 6 months ; and with the third, 4 years and 3 months. What was tlie servant's age on leaving his last master, supposing he was 20 years old on going to the first, and that he went directly from one to the otHer .? Ans. 34 years and 8 montlis. nJi' Ft^ "^^°y "^^y^ ^^^"^ *^« 3rd of March to the 23rd of June ? Ans. 112 days. 72. Add together 7 tons, the weight which a piece ot far 2 inches m diameter is capable of supporting • 3 tons, what a piece of iron one-thu-d of an inch 'in diameter will bear ; and 1000 Jb, which wiU bo sustained by a hempen rope of the same size. Ans. 10 tons, 8 cwt., 3 quarters, 20 ib. ' 73. Add together the following:— 2^., about the value of the Roman sestertius ; 7i^., that of the dena- rius; lid., a Greek obolus; 9d., a drachma: £3 15s A.d4^ 6s. 9rf., the Jewish talent. Ans. £bl\ 2s 74. Add together 2 dwt. 16 grains, the Greek drachma: 1 lb, 1 oz., 10 dwt., the mina ; 67 ib, 7 oz., 5 dwt., the talent. Ans. 68 ib, 8 oz., 17 dwt., 16 grains. QUESTIONS FOR THE PUPIL. 1. "What is the difierence between the simple and compound rules ? [8] . 2. Might the simple rules have been constructed so as to answer also for applicate numbers of difierent denominations.? [8]. 3. What is the rule for compound addition ? [9]. 4. How is compound addition proved > [16]. 5. How are we to act when the addends are numer- ous ? [12, &c.] tlili COMPOUND SUBTRACTION. jxins 120 acres, 3 ow much rood, 27 tlie first 2cond, 7 •s and 3 ving his )n going B to tllQ I to the a piece 'ting; 3 inch in istained tons, 8 »ut the 6 dena- '3 15s. el ; und achma; 7tj the ^ 123 ie and ted so fferent umer* COMPOUND SUPTRACTION. . 16. I^ULE--I. Place the digits of the subtrahend under those of the same denomination in the minuend-^ tarthmgs under farthings, units of pence under units of pence, tens ot pence under tens of pence, &c. II. Draw a separating line. * ™- ^"^^^act eacli denomination of the subtrahend trom that which corresponds to it in the minuend- begmnmg with the lowest. ,, ^7'J{ ^"^ denomination of the minuend is less than that of the subtrahend, which is to be taken from it, add to It one of the next higher— considered as an equi- valent number of the denomination to be increased : and, either suppose unity to be added to the next deno- mmation of the subtraliend, or to be subtracted from the next of the minuend. V. If there is a remainder after subtracting anr denommation of the subtrahend from the corro.?pond- lug one of the minuend, put it under the colmnn which produced it. yi. If in any denomination there is no remainder, put a cypher under it—unless nothing is left from any higher denomination. ^ 17. Example.— Subtract £56 13s. 4^,d., from £96 75. 6|c/. £> s. d. 96 7 6|, minuend. 5613 4|, subtrahend. 39 14 11, difference. We cannot take ^ from |, but— borrowing one of the pence, or 4 farthings we add^it to the I and then say 3 far! things from 5 and 2 farthings, or one halfpenny, remains • we set down i under the farthings. 4 pence ^om^/w« have borrowed one of the 6 peSce), anrone pcnny^Je naitpence ) 13 shillings cannot be taken from 7 but ^hor nC of /S^ ''T°- r ''* ^°^° 14 i^ «^e shillinl' place of the remainder. 6 pounds cannot be tak^n fmnf ^ K^y^ nave borrowed one of the 6 pounds in the minae^) $ 124 COMPOUND SUBTRACTION. })nt 6 from 15, and 9 remain: we put 9 under tlio units <.f one ol the 9), and 3 remam: we put a in tlio leas of pouud.i' place of the icuiamdiT. ^ «nnf; „T^*\^ """^^i ""*? ^^'^^ ""^"'""'^ ''f '* '^''« substantially the same as those already given for Siniple Subtraction [Kec. 11 J t, &c. J It 18 evidently not so necessary to put down cvDliera Where there ,s nothing in a denomination of the mlluff suiLetrnTscc. U^Ol ''''''' '" ''' ^^"^ "^^ ^ «-'P^° £ «. «?. From 1098 12 6 Take 434 15 8 663 18 10 KXERC1SK8. * s. d. 7G7 14 8 486 13 9 £ s. d. 70 15 6 14 5 £ «. d. 47 16 7 39 17 4 £ .?. 97 14 6 15 (> 7 From & s. 98 14 Take 77 15 ^ s. d. 47 14 6 38 19 9 * s. d. 97 10 6 88 17 7 £ s. d. 147 14 4 120 10 8 (10) £ s. d. 6()0 15 477 17 7 £ s. d. Prom 99 13 3 Take 47 16 7 « (12) £ s. d. *l^l 14 hh 476 74 (13) * *. rf. 891 14 li 677 15 61 (U) £ v. <;. 676 13 7^ 467 14 92 « (15) £ s. d. From 667 11 6^ Take 479 10 10^ (16) £ s. d. 971 Ok 7 (17) £ s. d. 437 15 11 14 (18) £ .V. d, 478 10 47 11 0^ (19) cwt. qrs. lb From 200 2 26 Take 99 8 15 100 II Avoirdupoise Weight. (20) (21) cwt. qrs. lb cwt. qrs. lb 275 2 15 9064 2 25 27 2 7 9074 27 (22) cwt. qrs. fl^ 654 476 3 5 f ft) from 554 fake 97 COMPOUND SUBTRACTION. Troy Weight. (23) (24) oz. dwt. gr. It) oz. dwt. gr. 9 19 4 946 10 16 15 17 23 125 (24) lb oz. dwt. gr. 917 14 9 798 18' 17 457 9 2 13 Wine Measure. (26) (27) (28) • (29) ts. hhds. gls. ts. hhds. gls. ts. hhds. gls. ts. hhds. gls. From 81 3 15 64 27 304 64 66 1 Take 29 2 26 3 42 100 3 51 27 2 25 2 52 Time. (80) (31) (32) yrs, ds. lis. ms, yrs. da. hs. ms. yrs. ds. hs. ras From 767 131 6 30 476 14 14 16 567 126 14 12 Take 4T6 110 14 14 160 16 13 17 400 15 291 20 16 16 33. A shopkeeper bought a piece of cloth containing 42 yards for iS22 105., of which he sells 27 yards for JS15 155. ; how many yards has he left, and what have they cost him ^ A7is. 15 yards ; and they cost him £6 15s. 34 A merchant bought 234 tons, 17 cwt., 1 quarter, 23 lb, and sold 147 tons, 18 cwt., 2 quarters, 24 lb ; how much remained unsold ? Ans. 86 tons, 18 cwt., 2 qrs. 27 lb. 35. If from a piece of cloth containing 496 yards, 3 quarters, and 3 nails, I cut 247 yards, 2 quarters, 2 nails, what is the length of the remainder } Ans. 249 yards, 1 quarter, 1 nail. 36. A field contains 769 acres, 3 roods, and 20 perches, of which 576 acres, 2 roods, 23 perches are tilled ; how much remains untilled ? Ans. 193 acres, 37 perches. 37. I owed my friend a bill of £76 16s. 9i-d. out of which I paid £od 17s. lO^d. ; how much remained due > Ans. £1Q 18s. 10-^d. n ■" 126 COMPOUND MULTli'LICATION. 38. A norchant bought 000 salt ox liidcs, wcigliln;; r>01 cwt.,2 lb; of which ho sold 2r)0 hides, Vei«,'hiu3 2.39 cwt., 3 qrs., 25 lb. How many hides had ho left, and what did they wci^^h ? Ans. 350 hidoH, woighin<r 321 cwt., 5 lb. 30. A merchant has 200 casks of butter, wcifdiing 400 cwt., 2 qrs., 14 lb; and ships off 173 c"ask,s, weighing 213 cwt., 2 qrs., 27 lb. How many casks has ho left; and what is their weiglit .> Anx. 36 cuska, weighing 186 cwt., 3 qrs., 15 lb. 40. What is the difference between 47 ]^]ngHsh miles, the length of the Claudia, a Roman aquoducjt, and 1000 feet, the length of that across tiie J^ee and Vale of Llangollen ? Ans. 247160 feet, or 46 miles, 4280 feet. 41. What is the difference between 980 feet, the width of the single arch of a wooden bridge erected at St Petersburg, and that over the Schuylkill, at Phila- delphia, 113 yarda and 1 foot in span .? Ans. 640 feet QUESTIONS FOR Til 12 PUPIL. 1. What is the rule for compound subtraction ? [16]. 2. How is compound subtraction proved ? [19]. COMPOUND MULTIPLICATION. 20. Since we cannot multiply pounds, &c., by pounds, &c., the multiplier must, in compound multiplication, be an abstract number. 21. When the multiplier doos not exceed 12 — Rule— I. Place the multiplier to the right hand side of the multiplicand, and beneath it. II. Put a separating line under both. III. Multiply each denomination of the multiplicand by the multiplier, beginning at the right hand side. IV. For every time the number required to mako one of the next denomination is contained in any pro- duct of the multiplier and a denomination of the multi- plicand, carry one to the next product, and set down the remainder (if there is any, after subtracting the number equivalent to what is carried) under the denomination i COMPOUND MULTIPUCATIOX. 127 wcigliln;* ho left, wuighing wciglung 3 casks, asks has 6 casks, >h miles, nd 1000 Vale of 280 foot, 'oet, tho cctcd at t Phila- 40 foot ? [16]. pounds, lication, t Land iplicand e. 3 mako ny pro- multi- )wn the nurnhfir liuation to whicjh it bolouf^s ; hut should then; bo lu) remaiudcr, put a cypnov in that duuouiination of the protluct. 22. Example.— Multiply £62 17s\ lOd. by 6. X 5. d. C2 17 10, multiplicand. 0, multiplier. 377 7 0, product. Six times 10 ponce are 60 ponce ; these aro equal to '> phillinpfs (5 times 12 ponce) to no carried, and no pence to be sot down in tho product — wo therefore write a cypher in tho pence place of the product. 6 times 7 aro 42 shillingM, and the 5 to be carried are 47 Bhillings— wo put down 7 in the units' place of shillings, and carry 4 tons of shiUings. 6 times 1 (ten shillings^ aro G (tons of 'lillings), and 4 (ten.s of shillings) to be carried, aro 10 (tons of shillings), or 5 pounds (5 times 2 tens of shillings) to bo carried, and nothing, (no ten of shillings) to bo set down. 6 times 2 pounds are 12, and 5 to be carried are 17 pounds— or 1 (ton pounds) to be carried, and 7 (units of pounds) to bo set down. 6 times 6 ('tens of pounds) are 30, and 1 to bo carried aro 37 (tens or pounds). 2o. The reasons of tho rule will be very easily understood from what we have already said [Sec. II. 41]. But since, in compouud multiplication, the value of tho multiplier has Jio cotinexion with its position in reference to tlic multiplicand, whore we set it down is a mere matter of convenience ; neither is it 80 necessary to put cyphers in the product in those deno- minations in which there aro no significant figures, " ^ it is in flimplo multiplication. 24. Compound multiplication may be proved by re- ducing tho product to its lowest denomination, dividing by the multiplier, and then reducing the quotient Example.— Multiply £4 3s. Sd. by 7. £> s. d. Proof : 4 3 8 29 5 8 7 20 29 5 8, product. 585 12 7 )7028 , product reduced. 12 )1004 io)ou e quotient reduced "~4 3 8=lnultiplicand. 123 COMPOUND MULTIPLICATION. pm«'i;„'trri''hUy performed" tr"' "•,"'«■"«'«, the »l.ouM be equal to ,T,e ICS ' " '°™"' '"'" "'' ">'' ihe quantities are to bo " rednr-pfl " hoft^r.« tu^A-- , - gnce^the^earuer i, .ot supp'^tit' be'S :%tT"a^iJe 1. 2. 3. 4. 6. 6. 7. 8. 9. s. 9 2 18 7 13 1 8 12 17 6 d. 3. 7i 6. li. U. Oi 0. 6. 6. 8. 0. EXERCISES. £ S. d. £ 76 14 7iX 2= 163 97 13 6iX 3= 293 77 10 74X 4= 310 96 11 7iX 6=482 77 14 6ix 6= 466 147 13 3^X 7=1033 428 12 7iX 8=3429 572 16 6 X 9=6155 ,A ii^ ^'^ ^ X 10=4288 10. 672 14 4 Xll=7399 W ir'l l^ ^ X12=9321 u u. 12. 7 lb at 5*. 2M #•, will cost £1 16s. Hd. 14* ?/l'ii' ""^ W^i*^- ^' ^"1 «<^«* ^4 18,. 5J,i. ii'J^ra\7ii3^t.^-^hrs^^^^^^^^^ Rule.— Multiply successively by its factors Example L— Multiply £47 13s. U. by 56 £ s. d. ^ ' 47 13 4 50=7x8 £ ,, ^ 333 13 4=47 13 4x7. 8 T 9 E Til [8ec, tlie 1 mult 2669 6 8=47 13 4x7x8, or 56. Example 2.— Multiply 14^. 2d. by 100. s. d. 14 2 100=10x10 ,. ,;. £7 1 8=14 2x10. 10 X70 16 8=14 2x10x10, or 100. COMPOUND MULTIPUCATION. ExAMPip: 3.— Multiply ^8 2s. 4c/. by 700. £> s. d. 8 2 4 10 — £ 129 81 3 4 =8 10 811 13 4 =8 7 s. d. 2 4x10. 2 4x10x10, or 100. 568113 4=8 2 4x10x10x7, or 700. The reason of this rule has been already given [See. II. 60]. 26. When tlie multiplier is the sum of composite numbers — Rule.— Multiply by each, and add the results. ExAMPLE.-^Multiply £3 Us. M. by 430. £ .9. d. 3 14 6 10 £ s. 37 5 x3=lll 15 10 d. £ s. 0, or 3 14 d. Cx30. 372 10 0x4=:H00 0, or 3 14 6x400. IGOI 15 0, or 3 14 6x430. r^'^^'V^nn" ?m *''^ ''""^^ ^'^ "'® «"'"° "« t^'^^t «li'e«<^y given [bee. II 52]. Ihe sum of the products of the multiplicand hv the parts of the multiplier, being equal to the product of the iuulti23hcand by the whole multiplier. EXERCISES. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. on 27. 28. £ 3 4 6 2 3 2 3 •s. 7 16 14 17 16 3 4 9 16 d. 6 X 7 X 6iX 6 X £ 18= GO 20= 96 22=125 36=103 66=214 64=139 81=261 100= 46 X 1000=816 100 yards at 9*. i^d. ^W, will cost £46 17 oVa S*i||^'*^ »' ^•^*- *'f- 4K, will cost 466 13 -4 gallons at (!.?. 8./. .W, will cost 80 bbO yards at 13a. 4c/. 4(^', will cost 240 X X X X s. 15 11 19 10 8 4 11 13 13 d. 0. 8. 11. 0. 8. 0. «.) ". 4. 4 6. 4. 0. 0. 130 COMPOUND MULTIPLICATION. 27 If the multiplier is not a composite nun^ci*--';-'^ lluLE. — Multiply successively by the factors of tho nearest composite, and add to or subtract from the pro- duct so many times the multiplicand as the assumed composite number is less, or greater than the giv9i-\ multiplier. Example 1 —Multiply £G2 12s. Od. by 70. £ s. (L 62 12 6 8 76=8x9+4 501 9 £ s. (I. 4509 0=G2 12 0x8x9, or 72. 250 10 0=02 12 0x4. 4759 10 0=62 12 0x8x9+4, or 70. Example 2.— Multiply £42 3s. 4(L by 27. £ s. (I. 42 3 4 4 27=4x7-1 108 13 4 £ s. 1180 13 4=42 3 42 3 4=42 3 d. 4x4x7, or 28. 4x1. 1138 10 0=42 3 4x4x7-1, or 27. The reason of the rule ia the same as that already given [Sec. II. 61]. ^ ^ EXERCISES. £ s. d. £ s. d. 29. 12 2 4 X 83= 1005 13 8. 80. 15 0ixl46= 2193 3 Oi 31. 122 6 X102= 12469 10 0. 82. 963 0^X999—962040 2 5i. 2-8. When the multiplier is large, we may often con- reniently proceed as follows — Rule.— Write once, ten times, Sic, the multiplicand, &c., of the multiplier, add tho results. COMPOUND MULTIPLICATIO^f. 131 Example.— Multiply £47 16s. 2d. by 5783. 5783=5 X 1000 + 7 X 100+8 x 10 + 3 x 1. £> •«. d. £ s. d. Units of the multiplicand, 47 10 2x3= 143 8 6 10 Tens of the multiplicand, 478 1 8x8 = 10 Hundreds of the multiplicand, 4780 16 8 X 7 = 10 3824 13 4. 33465 16 8. Tliousands of the multii.Iicand, 47808 6 8x5 = 239041 13 4. Product of multiplicand and multiplier =27647.5 11 10. EXERCISES. 33. 76 14 84. 974 14 85. 780 17 d. £ .9. (I. 4 X 92= 7057 18 8. 2 X 76 = 74077 16 8. 4 X 92=71889 14 8. 7ix 122= 9013 10 3. 7ix 162= 6865 11 lOJ. 38. 76 gallon.^ at £0 13 4 4f , will cost £50 13 39, 92 gallons at 14 2 4f , will cost 65 3 36 37 73 17 42 7 4. 4. 40. What is the difference between the price of 743 ounces of gold at £3 17s. lO^d. per oz. Troy, and that of the same weight of silver at 62d. per oz. .? Ans. £2701 2s. 3^d. 41. In the time of King Jolm (money being then more valuable than at present) the price, per day, of a cart with three horses was fixed at 1^. 2d. ; what would be the hu-e of such a cart for 272 days ? Ans. £15 175. 4d. 42. Veils have been made of the silk of caterpillars, a square yard of which would weigh about 4 grains ; what would be the weight of so many square yards of this texture as would cover a square English mile > Ans. 2151 tb, 1 oz., 6 dwt., IC grs., Troy. QUESTIONS TO BE ANSWERED 13Y THE PUPIL. 1. Can the multiplier bo an applicato number ? [20J. 2. What is the rule for compound multiplication when the multiplier does not exceed 12 ? [211. 3. What is the rule when it exceeds ^12^ and is a composite number ? [25] . m i!li 132 COMPOUND DIVISION. 4. When it is the feum of composite numbers ? [261 6 When It exceeds 12, and not a composite number ? 6. How is compound multiplication proved ? [24]. COMPOUND DIVISION. 29. Compound Division enables us, if we divide an apphcate number mto any number of equal parts, to asceitam what each of them will be; or to find out anrth'^r^"^ ^'"'''^ """^ applicate number is contained in If the divisor be an applicate, the quotient will bo an abstract number— fur the quotient, when multiplied by the divisor, must give the dividend [Sec. II 7yl • but ronV^^^ri"!!' ?>'^^''' ''^"'"^* ^« multiplied together L20J. If the divisor be abstract, the quotient will be applicate— for, multiplied by the quotient, it must give *he dividend-an applicate number. Therefore, either tlivisor or quotient must be abstract. ceeTl^^^"^ *^'^ ^^^'^'^'' ^^ ''^^*''^«^' ^^^ <ioes not ex- RuLE— I. Set down the dividend, divisor, and sepa- rating Ime— as directed in simple division [Sec. II. 72]. II. Divide the divisor, successively, into all the deno- minations of the dividend, beginning with the highest 111 Put the number expressing how often thl divisor s contamed in each denomination of the dividend under that denomination— and in the quotient tinnTf tL* r •^''''',°' ''^'^^J contained in a denomina- tion of the dividend, multiply that denomination by tho number which expresses how many of the next lower denommation is contained in one of its units, and Tdd the product to that next, lower in the dividend. wn V * ..A^^^^l ^""^^ succeeding remainder in the same way, and add the product to the next lower denomi- nation in the dividend. umomi VI. If any thing is left aftor thn nnoti-nt ^"o-- -^i- lowest denomination of the dividend is obtained; pi^t'iJ COMPOUND DIVISION. 13: in I down, with the divisor under it, and a separating lino between : — or omit it, and if it is not less than half the divisor, add unity to tho lowest denomination of tlie quotient. '61. Example 1.— Divide X72 6s. did. by 5. £> s. d. ' 5)72 6 14 9 4i 5 will go into 7 (tens of pounds) once (ten times), ami leave 2 tens. 5 will go into 22 (units of pounds) 4 tiraos. and leave two pounds or 405. 405. and (Ss. are 4Gs., into which 5 will go times, and leave one shilling, or lid. 12t/. and \)d. are 21tZ., into which 5 will go 4 times, and leave Ir/., or 4 farthings. 4 farthings and 2 farthings are 6 farthings, into which 5 will go once, and leave 1 farthing— still to be divided ; this would give \, or the fifth part of a farthing as quotient, which, being less than half the divisor, may be neglected. A knowledge of fractions will hereafter enable us to understand better the nature of these remainders. Example 2.— Divide £52 4s. l^d. by 7. & s. d. 7)52 4 n 7 9 2 One shilling or \2d. are left after dividing the shillings, which, with the Id. already in the dividend, make 18(7. 7 goes into 13 once, and leaves 6rf., or 24 farthings, which, with f , make 27 farthings. 7 goes into 27 3 times and G over ; but as G is more than the half of 7, it may be consi- dered, with but little inaccuracy, as 7— which will add one farthing to the quotient, making it 4 farthings, or one to be added to the pence. 32. This rule, and the reasons of it, are substantially the same as those already given [Sec. II. 72 and 77]. The remain- der, after dividing the farthings, may, from its insignificance, be neglected, if it is not greater than half the divisor. If it is greater, it is evidently more accurate to consider it as giving one farthing to the quotient, than 0, and therefore it is proper to add a farthing to the quotient. If it is exactly half tho divisor, we may consider it as equal either to the divisor, or 0. 33. Compound division may be proved by multipli- cation — since the product of the quotient and divisor, plus the remainder, ought to bo equal to the dividend [Sec. TI. 79]. g2 134 COMPOUND DIVISION. EXERCrsKS. 1. o 90 7 7(5 14 47 17 10 11 3. 4. 6. 6. 7. 8. 97 14 9. 147 14 167 16 176 14 96 19 77 10 32 12 44 16 d. £ 6-1. 2=48 7-j- y=25 0-^ 4=U 4-:- 5=19 0=12 7=4 7-i- o ft. d. 3 9. 11 6i. 19 U. 7 10.^. 19 5i. 13 '3. 7 -f. 8= 5 12 l" 17 1|. 16 5i 3-^ 9=10 0-4-10=14 7-Ml=14 6-j-12=14 6 14 11) 6i The above quotients are true to the nearest fa. dnl^. number!?''' '^' ^^"^^''^ "^'^^^^ ^^' ^"^ ^^ ^ compos-it. Rule.— Divide successively by the factors. ExAMPLt.-Divide £12 175. U. by 36 3)12 17 9 This rule will be understood from Sec. II 97. 12. EXERCISES £> i. d. £ ^. J* II 6-^ 24= 1 d. 81 13. 676 13 34- 36=16 4] 14. 447 12 2-5. 48= 9 6 6 16. 647 12 4-^ 60= 9 i? 7 17. 740 13 4-5- 49=15 2 sj. Jte n^W- ^^"^ '^'"''^ 12, and ia no. a oom- h„fi!i!''^7^™'"""^, ""y *''<> ""ttod of Ions division • set down tlie inultiplic-a, fco obt-inM "' «"'aond, tient as directed in long'dH.ision 'tSo II.'S]! '^'"" ■"^ iposite lom- lon ; tiers eno- iiext [)nd, j[uo- t;OMP0U.\D DIVISION. 135 ExAMPLK.— Divide £87 IG.v. 4d. hy G2. £> s. (I. £, ,<. f/. 62)87 10 4 (1 8 4. G2 25 20 multiplier, shillings 5 16 (=25x204-10) 490 "20 12 niultiplior. pence 244(=20xl2-f4) 186 ~58 4 multiplier farthings 232 (=58x4) 180 "40 C2 goes into £87 once (that is, if gives £1 in the nuoti.nir), ard leaves £25. £25 are equal to 500.s\ (25x20j, Avhich, with 10.S. in the dividend, make SlCs. 02 goes into 51fis'. 8 times (that is, it gives 85. in tlie quotient), and leaves 2(»i-., ov 240^2. (20x12) as remainder. 02 goes into 240, &:c. Were avo to put I in the quotient, the remainder would ba 40, which is more than half the divisor; we consider tlm quotioiit, therefore, as 4 farthings, that is, we add one penn^ to (3) the pence supposed to be already in the quotient. £1 8*'. Ad. is nearer to the true quotieut than £1 8s. 34Vi.[32]. This is the same in principle as tlie rnle given above [30]— buv since the numbers are large, it is more convenient actually to set down the suras of the dilVorent denominations of the divi- dend and the preceding remainders (reduced), the products oJ the divisor and quotients, and the numbers bv which we multi- ply for the necessary reductions : this preveuta the memory from being too much burdened [Sec. II. 93]. 36. When the divisor and dividend are both applieate numbers of one and the same denomination and no reduction is required — FtULE. — rroeeud as already dii'ected fScc. II 70, 72, or SU]. ' 136 COMPOUND DIVISION. - Example.— Divido £45 by £5 £5)45 Tliat is £5 is the ninth part of £45 nominatiou is found in cither, oi' both^ '"' ^'• ItULE.--Ileduco both divisor and dividend M fl.n i est denomination contained in eithor m / .u^ ^^"^ ceed with the division ^^^' ^^^ *^^^^ P^^ ExAMPLE.--Divide £37 5.. 9ld. by 3.. Ghl '' i'. £ s. d. ' ' 3 12 "42 4 170 farthings. 5 9J M 170)35797(211 340 179 170 "~97 Theroforo 3.. Qid. i, (ho 211th part of je37 5y.9,V/. *J7 not being less than the half of 170 ^91 ^^^ -i . OS equal to the divisor, and therefore addSf J fT Tr''^?'' '* as the last quotient. ^"^^eioie add 1 to the obtained 18. 19. 20. 21. 22. 2-3. 24. 25. 26. 27. 28 29 SO £ s. 176 12 134 17 1736 14 73 16 147 14 157 IQ 68 15 62 10 8764 4 4728 1;} 8204 5236 2 EXEHCISES. d. 2 -^ 8 ~ 7 -^ 7 -f- s -^ 7 -1- 2 -i. 6i^ Oi~ 2 -f- H-i- 7|-^ 2 —9842: • 191: 183: 443= 271: 973= 487= 751 = 419= 408= 317= 261:= 875= £ s. ■ 18 d. 6. 14 9. 10 13 104 " 6 3 6 1 2 5i. 5|. = = = : :18 14 :14 18 31 10 114. ■■ 5 19 81 9 4]. 111. Gh. 41 COMPOUND DIVISION. 137 licato, but a one de- ) the loT? then pn. 31 . A cubic foot of distilled water weighs 1000 ounces what will be the weight of one cubic inch ? Ans 253'lS2i) grains, nearly. 32. How many Sabbath days' journeys (each 1155 yards) in the Jewish days' journey, which was equal to ii3 miles and 2 furlongs English > Ans. 50-66, &c. 33. How many pounds of butter at ll^d. per lb <vould purchase" a cow, the price of which is £14 15s. ? Am. 301-2766. QUESTIONS FOR THE PUPIL. 1. What is the use of compound division .? [29]. 2. What kind is the quotient when the divisor is an a])stract, and what kind is it when the divisor is au applicate number ? [29] . 3. What are the rules when the divisor is abstract, and docs not exceed 12 ? [30] ; 4. When it exceeds 12, and is composite ? [34J ; 5. When it exceeds 12, and is not composite ? [35] ; 6. And when the divisor is au applicate number ? [36 and 37]-. '■ i^ lh( isidor it )btainoJ 13S Sii:oTioN jy. FRACTIONS. ?n<l one TZZ'^CtZlf^'f.''''" ^I""' P"'«. is called a /„„«« ""''° P«'« »'•'> token, we have what re,t';4:r;';:;'Lt3e-St'%'''!,.''--- "- been la a fraction of 5— its siv(l, 1 . !i •'^ '' f^""- H- ««!) into six e,,„al part f w n ?tn ' *""" ""'Z boingdivid i wo shall sJ-o prLen iy) »' un?^^™?,^ *'""» i "' (»» 4 Srchtgo^fcl^ZV"^^^^^^^^^^ » fr- or, while the Jeuom „'uoV tell the ^' ''"■mimtor ,■ kinci of parts into which the nn;,- *"'"'"««*« or divided, the numerator "m„!t/« J, '"^f"""^ '» ""o number of them which TtTn t!"' "l f*'"""'^ 'l-^ sevenths) means thottL ! ,/''"M (read three- " 'tree- V"dr'ttXm''ed ^^r''^'" ""^ "-' 'Trei;::?;t:'/» ^~th?f::r:.-» -^ of the fraot! nX~h™ .'• '^^ ?'°'" ""^ ™'" JiviJc the numeSor bv thT"J'™' "^'"^'^ *'"=" we value ; and the Kreater .l,r^^ •T"]"'"''''' '' "« ■''^al quotient. On the^eomrart .h7*",'* '¥ ''■"■§«"■ «'o ♦or the less the frae on^:,-' T''^ ""^ denomina- '1- smaller the quTt „t7seo jf 7^^^^^^^ "^"-^ greater than 4— whieh ;« i? i T®^ =— hence 4 is nun.r'atrr^ ?dXls^r' V'''^^-«-' "' »' ^l-«-'i-neha,^^d.1l^X;r^lSn"-S • -^ xro ean no FRACTIONS. 139 to increaso or <liinini«U both the (Ji\ iJciiJ and divisor — which does not afFoct the (diotieut. 5. The following will rc^. resent unity, seven-sevenths, and live-sevenths. I I U"^^y- I Lii ?i The very faint lines indicate wliat ^ wants to make it equal to unity, and idmlical with ^. In the diagrams which are to follow, we shall, in this manner, generally subjoin the difference between the fraction and unity. The teacher should impress on tlie mind of the pupil that he might have chosen any other unity to exemplify the nature of a fraction. 6. The following will show that ^ may be considered as either the -f of 1, or the | of 5, both — though not identical — being perfectly equal. \ of 5 units. ? of 1 unit. Unity, DIDIl m 1 \fi\f o\ Tn the one case we may suppose that the five parts belong to but one unit ; in the other, that each of the five belongs to differeii ' units of the same kind. ^ Lastly, ^ may be considered as the | of one unit fivo times ua large as the former ; thus- I of 1 unit. I of 5 units. equal to ■I .. .1 . -. 0\ "•* HO FnACTIONS. at leapt, of the quotiont^nUo'Llntotr '"'''''•' ''^''' not. uknHcalZk''Z , j^lf C.S'T*"V!"' ^' lowmff will exomnlifv +].! • "'^^"^" h iho foi- thus J i3 evidently eqS to 'r"*' *f f »''■">»"*»■■ ; noticed when we trite'd of divisfo^ Lril.Tl/'"^"^ tbo>;e4eteabt";7t\"n;tL"° '"''™ ".^Z^'' KKACTION«. - I4j 11. A cnmpnwnd fraction siipposos ono fraotioa (o refer to another ; thus J of J— roproscntiid iilso hy J X ^i (throe-fourths multiplied by four-ninths), means not tho four-ninths of unity, but the four-nintlia of tho throo-fourths of unity :— that is, unity boia,^' divided into fjur parts, tlirco of thcso arc to bo divided into nino parts, and then four of those nino arc to bo taken ; thua I ? o TIT I r 12. A complex fraction has a fraction, or a mixed number in its numerator, denominator, or both : thus t, 4 which means that wo are to take the fourth part, not of unity, but of the ^ of unity. Tiiis will bo eicm- pufied by — 8 ?- U U - 3 r 1 f<:cj^ J 1 ^ u 7' s' 1 -4 5 — , are complex fractions, and will be better 54- understood when we treat of the division of fractions 13. Fractions are also distinguished by the nature of thou' donommators. When the denominator is uniiv, foUowed by one or more cyphers, it is a decimal frac- iion—imL^, /it, f Ao , &c. ; all other fractions are vilo-ar ^thus,i,|,73_, &c. Arithmetical processes may often be performed with fractions, without adnalhj dividing the numerators by the denominators. Since a fraction, like an integer may be increased or diminished, it is capable of uJli* tion, subtraction, &c. 142 FRACTIONS niinaticm!' '''^'^' """ "'^'°"' ^° '' ^''''^^^'" '^" ^^^ ^^"«- uni^v"" !r.^T' "^^ ^ co^'^idercd as a fraction if wc make unity it^ denominator .-—thus f may be taieu for 6 • if 21° '""^ ^7" ™, '."J^S"!- any denominator we please If we previously mnltiply it by that denominator ; tims, 5=1 , or f , or f , &c., for 22=5X5_5_5 . , 30 5X6 5 ' ^ 1X5-1-5, and -— = =-=5 &« 6 IXQ I ^ EXERCISES. ^i S^'^''''^ '^ *° ^ ^'■''''*'''°' ^^""'"S ^ ^' denominator tor^" I'f."' o^^ *° ^ ^'^'*^'"' ^^"^^"S ^^ ^« denomina- ^',i^^'' l^^'- 1?=V. I 5. 42= V/. I 6. 71 = «F« lo. lo reduce fractions to lower terms Before the addition, &c., of fractions, it will be often Trirplpe- ^^ *'^^^ *^^"^^ ^^ ^^^<^^ - P-^^^- ■n, 40 5 40 40-i-8 <! Example.— =j^=-. For ~— 3^^Xr_^ 72 9 '^^'72-72-^8-9• We have already seen that we do not alter the quotient- which 18 the real value of the fraction [4]-if we XSv ot divide the numerator and denominatoh)y the Tame number he^atr:' ^''" ''''' ^''- "• ''"' ^^^^ ^^ usefull^rem , : Reduce the 7 -574 387 10. MJ% 12 ?§CFr' In the answers to in future, generally mlnations. EXERCISES. following to their lowest terms. 13. 14. 15. 16. 17. 18. Fa 3 1 • 4 3 4 Fll 3"* 60 5 12 S"- .98 7 1 1 2 — y 19, 20, 21. 22. 23. 24. 100400 1004 7^00 2' 5UO0 1120 firnj — t:?ii' 4 2 5__ s's" ' 1^^, — I?T' 30 8 4 12. -"BTr="-2n-3 5I2_2 5 B J i — SoT questions given aa exercises, we shall, reduce fractiona in fhoJi. In^nof d Z_ FRACTION*. 143 ny deno- we make I for 5; e please 5 nmator lomina- 8514 »4 ' 16 often ossible, ommon tient— tiply or umber, •emein- — 1004 -3ffoa- 1 20 ?T1' h' shall, ^ — 28. £-'=ir).v. 29. £-j^-=5.s-. 30. £^|„=1./. 1«.'|^ find the value of a fiactiou in terms of a lower a6iaomination — RuLR.^;^rteduce the numerator by the rule already given [Sec. III. 3], and place the denominator under it. ExAMPLK.— What is the value, in shilUngs, of J of a pound ^ £3 reduced to 8hillings=60.s. ; therefore £'i reduced to shil- liiigs=';j^s. TJie reason of the rule is the same as that already given [Sec. III. 4]. The | of a pound becomes 20 times as much if the " unit of comparison" is changed from a pound to a shilling. We may, if we please, obtain the value of the result- ing fraction by actually performing the division [91 • thus \°s.~los. :— hence £^ = l5s. * EXERCISK.S, 25. £U=Us. 6d. 2G. £1^=1 7s. 4d. 27. ^ll-^lOs. 17. To express one quantity as the fraction of an- other — KuLE. — Ecduce both quantities to tho lowest deno- mination contained in either — if they are not already of the same denomination ; and then put that which is to be the fraction of the other as numerator, and tho remaining quantity as denominator. Example.— What fraction of a pound is 2W. ? £1=9G0 earthings, and 2,|./.=9 farthings; therefore ^fr^ is the re- quired fraction, that is, 2^(1. =£y^^. om '^•^Tt^ ^*' "^^""^ lluLE.— One pound, for example, contains JfM fartlnngs, therefore one farthing is £-.' (the OGOth part of a pound), and 9 times this, or 21, is £9 X y|o=ffT!r- EXERCISES. 31. What fraction of a pound is 14?. 6d. ? Ans. ^a 32. What fraction of £100 is 17^. 4d. ? Ans. -ia°." 33. What fraction of i^lOO is i:32 10^. ? Jm-.*i|! 34. What fraction of 9 yards, 2 quarters is 7 yar*ds, 3 quarters ? Ans. .ai. *^ ' 35. What part of an Irish is an li^nglish mile > Ans n 36. What fraction of 6s. iid. is 2s. Id. ? A,is. X ' ov. What part of a pound avoirdupoise is a pound Troy? Am. -m. ^ u^ «.' VULGAR FRACTIONS QUESTIONS. 1. What is a fraction ? [1] .. 7/ >Vhj may the numerator and denomhiatnr he. «,„i o. What IS an improper fraction ? [71 b. What IS a mixed number ? [8] numbejTrg]"" ""^^P^' ^'■^'"^"^ '"^uccd to a maed denomination? [14]. ^ *^ "^ fraction of any ^^12. How is a fraction reduced to a lower term? a ^Jt2^^^^ ^-^- ^-^^ - terms of of ItotfeT' tl7T ''^'''' '"' ^"'"'^^^ ^« the fraction VULGAR FRACTfOiNS. ADDITION. 18. If the fractions to ho Tddo/I i,n,,^ denominator— ^^ ^^'^^° ^ common Example. — a 4. g — i t Reason of THF- Tfrrrw T^ -I, if VULGAR FRACTIONS. 145 tbeir nature. Unity. 1. 3-K+^=v==i5. 3. |.'4-'"4.iL=;!o_o4 O. f ^-U'-*_l_lo 4 2 03 EXEnciSES. I t), l^J_ll_l_l«_33_0 10. -lT4.'t!-f.iii_.5_^_.oin 12. ^\-f a 4.11^3^^1 ri? 19. If tlie fractions to bo added Iiave not a'c'ornni..n a|«ator, and all the denonnnato.s are p iL^rS Example.— What is the sum of 5-f 34.4 ? |+^+ ^=.2X4X7 3x3x_7 4x3xi ^6 63 48 m H • ;^t^'^X^X^^7x3x4=84+84+84=8T factors^ (tlio ' v^Mdenomnnt T"' "''/"'' ^ ''^"«« ^'^« ^=""« the same product '^'"""^^'^^^^^ors) mu«t uecemrilj produce confmof trmrn'aoT we^'w""''' ^^"^ "'^ ^^^^--'^ *« a rator and denomi air of e^^ "Ct/^^'^'P"''^ ."'« ""'"«' [4] does not alter the fraction "^ Tf- '^'"^ ""•"^'''•' ^^'^<^^ common denominator; for 7f L „ ],/%r''T'''-^: ^^ ^""^ <* out so doin,, w. cannot pil ^1^ tloililaJTri^-^n^it of them as tlie denominator of tlieir sum;-thus ^^±^+^ So ^uSffi^r t jiii^ns^r .:? r ^ -^^'^ ' ^^^ and sevenths. Avhich avo Lo! 1, J'"''P."'^ ^''^"» "»« <onr(l,3 N-^+i are less t) be correct — tl.an thirds; noiihor wuukt «ince it would suppose nJi ,<y tU«m to be f 146 ! 1 Tilt , lis VULGAR FRACTIONS. equivalent number of others wl.icl L"e ,„,„ll,r 1 ' '° "° H+?=2_X4 3X3__8 9 17 _ 3 4 8X4'^4X3~12+12"=J2 These fractions, before and after thev r..r.fMvn .. denommator, will be represented as follo'y^s :- '"""'^ 8 1 (' equal to ■ w iai n an^JL *— as w: ilr diss ihd/^i^^^f °' ^'^^ r? j-^ - --'« than twelfths, we eouk not C« f ''!,"*'^ ^"'^'" I^'"'^'^ l'"-^'««- exactl,oquivalenl"res;!ectitely:iot?AS^r '' '''"' 12- 3+f+f EXERCISES. 20 =5 9 139 22. If the denomiuutor, to each other- Procoed as Rule. — Fi denominators fractions to be added have not a common and all the deuommators are not primo directed by the hist rule ; or— "d the least common multiple of all the [Soc.II.l07,&c.|,thiswilllieth.cnnnnnn muifciply tno numerator of each fraction 8n_i 5 i coniiiiou VULGAR FRACTIONS. ^4-* into the quotient obtained on dividing the common mul- tiple by Its donominator-this will give the new nume- vators ; then add the numerators as already directed [18]. ExAiMPLE. — Add X -I- A -I- 3_ OQQ • .1 , ■^^^ 3 2 1-4^5 -f-^ 2- Z6ii IS the least common multiple of 82, 48, and 72 ; therefore --i.l.i_288-i-82x5 .281-f:48X4 288^72X8 45 24 ' if 8^"""^^^"" ii88 + 288 =288+2T8+288=2T8- multiple of the denomiLts^K^Tnt ^6^si'JT'''' __ 5><288 ^ ^^""^ instance)- g^q-^. For we obtain the same quotient, whether diminish the numher^f^^^^^^^^^^ tionsTe oS.lr h^'''''^'"' ""? ""'^ P""^° *'' «^«I^ other the frac jS>?e ff ^e a:::^:z:::^i^j^t^]^^-:^ denommators, the common denominator irthe present n stance, had wo proceeded according to tho L, ru^e ?9], wi would have found 1 , j. , j. __ 17280 18432 4608 4032£ . ^^^ 40320 ^^^^"^2-110592+110592+110692=* 110692 ■ " 110592 ^^ evidently a fraction containing larger terms than --. 288" 25. ^_1_SI4.^14 3 O'JS 2b. 44.^_Lj__t;7 9." 27 ?TfTf >2— -r2.- EXERCISES. ^4-5 i 4_.:jii.3__2-4 7 34. ;v?|+|=fCl^ 35. |flflL45i?'' 37. |+HJ!f!fe:Z:?;T^7 »2T0"' 28. l+l-Li^^dllZ^Vl^- 29. iJiJ niiST^^- 30. |I|t^=,i?Z{y tioni^'" ''^'''" ' ""^^'^ ^^"^^^^ to"iu" improper frac- ha^n.''';il^!?"^%*^° '^^'^'''^ ^'^'^ i«i« a fraction, te.^:dn;tri^L.^,.r ^-^-^ p- ^^Ex^PLK.-What fraction is equal to4f ? 4^ = 4_f.5=, Mi II '. 148 VfLUAU FRACTIONS. -o. Reason- of thk Rirr p _ \Va i „ i •. integer may bo expre;sc. us a ^r u Hon f '"'''-^ '''^ '^'^' ^ nator >ve pksase :-the reduction nf„ •'"'T'''" ''"^ ^^'"^""- fore, is really the addition of Z.r '"'''''' ""mhor, tliere- a coiiiiuou deuomiaator. ^^''^"tions, previously reduced to EXERCISES. 44. 99A.=:iono 45. 12''.i=>i^ • 40. i4Lo/: ■ 47. 40 -'==3 73 48. 13 =1 o; 49. 27||=iv. 38. 16'=''3^ 39. 18|=i|o. 40. 79.=«|3. 41. 47 ='|3. 42. 741=017. 43. 95^=^f6. 26. To add mixed numbers— KuLE.--Add together the fractional parts- th^n if «ho sura IS an imnroner fr-^ot;..^ T P. ^^' ^'len, if ■="»"■« 1-What is the sum of 41 + 18 " 7 I 5 12 14 «T^8 — 8 =J-i sum isl eighths-that ; , one o^t a;Hed''^n;: .**; T '"''' ^'^ ^ and 18 are 19, aid 4 are 23 ' "" "'*^ ^"'^"- -^ KsAMPLE 2.— Add 12f and 29i.l. |4-|-|==4l=l.i 7 12^=12.3 5 3 sum 42'-I y T„ .1 • . . sum 4z.'ri rW an,r 22f t'XTS l^fj H'''""""*? ""^ -Edition aenominatol^ ""* ''^"'"""^ l^to to ,v oomiuon beS-is"ptSir,?er™'il"-*|"'» ^' "'-0 '-™- but, in the first exa™t.efi,ril?''''? f ""P'" "M'ion Won is equal to „S oTthl „" . ST'-"*^' ''.'' O"" <>™"mi„a. .1. 8], J« Of one dealttr„'^*!S '^ TeTf ttt"." ''^° 50. «+3|=8S. '"^"'-■'"='- 61. 8f4.4.2|l==lltRi 52. l65'H-fOG^L^"' 53. 10+11^.^22^ 54. II MS 56. 4r+3§j:H6 =n |»- 58. 92.4.+37Ji+7t-ifi7J5, 59. 17aA+8!3tl|T=27^||J. VULGAR FRACTIONS. 149 th.it an dcnomi- '', tliere- (iuced to if set QUESTIONS. 1. Wliat is tho rule for adding fractions which have a common denominator? f"18]. 2. liow are fractious brought to a common denomi- nator ? [19 and 22]. 1 ^' S** ^^ *^^ ^"^^ ^^^' addition when the fractions have different denominators, all prime to each other ? |_lc)J. 4. What is tho rule when tho denominators are not the same, but are not all prime to each other ? [22]. 5. How is a mixed number reduced to an improper fraction ? [24] . ^ 6. How are mixed numbers added ? [26]. SUBTRACTION. 28. To subtract fractions, when thcv have a common ionommator — ^ .Rule.— Subtract the numerator of the subtrahend ifrom that of the minuend, and place the common deno>. iftimator under the difference. Example.— Subtract -J from J. 7_4_7--4_3 9 9~ 9 ""9' 20 IlEAsoN OF THE lluLE.— If We takc 4 individuals of anv Kind, Irom t of the same kind, three of them will remain In the example, we take 4 (ninths) from 7 (ninths), and 3 are loft— wluch must be nintlis, since the pvoces.s of subtraction cannot have changed their nature. The fullowin- will exempUfy the Bubtractiou of fractions :— a ^ ni>uiy me ^^ Unity. 7 IT w r 73 4 iiii i a> I H Uil'i' I II i'lr Ihif 160 VULGAR FRACnONS. 1. 2. 3. 4. 5. EXERCISEa. — ^,=t.. c. 7. 8. 9. 10, 1* « . 5 U-XAMPLE.—Subtract ^ from |. ft! T> " " -^2 i"5=^tf. OX. JtRASOIV OF THP T?tttw tx • g;^ven f20] for reducing fractbns !« !''"''''' *" *'^'*^* ^^^^^'^^^ previous!; to adding them ^'^ "^ '^^'^'^o" deaominator'. 11. f- 14. Il-i^ll^s '* 13 — Iffy. EXERCISES. _S 7 _S — 39 "8' I 15. 16. 17. 18. 11^^13X_ 769 11-4 HZ's'^^** 4 8 8i — -o-g. 75H_320_2;! m^fd n'umberf'' '"^^^ "^°^^^-' ^^ frrctions from iVitfrsXL^^^^^^^^^^^ :.---" denominator- W from that of the ifnd 1'.'' ^f.^ ^' *^^ ««^*^- ence with the commS^ Zn • /'* ^^^^ *^« ^^^ff^r- subtract the integTpart ofT k'. T^'.' ^'^ *^^" integral part of the minueL subtrahend from the that of'Ih^:/£t nd, Clt r r '' ^- *^- mon denominator to ts nu2r./ ^ .'^'"^ *^*^^ «^°^- mtegralpartofthemi^uendT;^^^^^^^ '"' '^^^''-^^ *^' Example 1.~-4| from 9|. 9f minuend. 4| subtrahend. ^ /»• vxi. y. ^*^ difference. from &'^S,4!'«''"'' -d 2 eighth. (=.; ,,„^, , .VULGAR FRACTIONS. 151 not a coin- inator [19 rule. at already uominator. 7_e9 2" ns from inator — subtra- e differ- !; : then •om the ss than com- ise th'i ka. 4 ExAMPLK 2.— Subtract 12.^ from 18]. 18j minuend. 12^ subtrahend. 5^ difference. 3 fourths cannot be taken from 1 fmirth ; but (borrowing one from the next denomination, considering it as 4 fourths, and adding it to the 1 fourth) 3 fourths from 5 fourths and 2 fourths (== j) remain. 12 from 17, and 5 remain. If the minuend is an integer, it may be considered as a mixed number, and brought under the rule. Example 3.— Subtract 3f from 17. •,JJ i!?^^y ^® supposed equal to 17^; therefore 17-34=3 17^-3^. But, by the rule, 17^-3J=16|-3f =13^. 83. Reason of the Rule.— The principle of this rule is the same as that already given for simple subtraction [Seo 11. ly] :— but m example 3, for iustance, five of one denomina- tion make one of the next, while in simple subtraction ten of one, make 07ie of the next denomination. 34. If the fractional parts have not a common deno- minator — Rule. — Bring them to a common denominator, and then proceed as du-ected in the last rule. Example 1.— Subtract 42| from 56^. 56|==56y*^, minuend. subtrahend. 42 > = -42X I'lyV' dilTerence. 85. Reason of the Rule.— We are to subtract the dif- lerent denommations of the subtrahend from those which cor- respond m the minuend [See. II. 19]-but we cannot subtract iractions unless they have a common denominator [30]. EXERCISES. 19. .20. 21. 22. 23. 24 15|-7|=7 12f— 12 8411 _lt_; 941 14iif— |f=1473-. 24. 82iH-7iif=74e. 25. 762-72/^^3 Jf."^ 26. 67|-34X=32^1. 27. 971-32J|=64-TC 28. 60|-4ll(=19i! 1001— 9|=9ni' 60— A=59,« '■ 29. 30. 31. 32. 12|-l01= t u-^ i'l y I6d TULQAR FRACnONS. .hi I" '^ |i QUESTIONS. <^ 9 W»,o* • ^u ^^'""^on (ienouiinator ? [281 MULTIPMCATION. tholonLy^""'^ * '■'•''°"»'' l-y » ''hole number; or ExAMPLE.—Multiply f by 5. 37. ReASOIV of the TfTTn? T 1 . we are to add the multS^ca^ r«-" T?^^oi*^ ^^^ ""'"ber. as are indicated by the mu t?pSer?^but ILf?/' V^^^^y times a common denominator we must add ftn ""^^ factions luaving put the common denominatorTn'dfk^o'^rX^^^^^ -^ whTchcoSSe'tL^Scin'^o'^^^^^ "P^^^'«"'«f *hV integer multiplier-their s"zeSTunoZZ"''n '^P^?f "^ ^^ *h« be the same thing to incrlR7fh!?l^ • l^ ^^^^^ evidently Without altering %l,eirS'^!^S,lT ', \" ^^"'^l «^t««t dividing the denominator bvTh?l-'^°"^'^ ^^ ^^^^t^d by AX5 = -|. This will become Jll. ^'''° -i^^^^^P"^^ 5 thus the fractions resulting W Uf C£^t^o^otfr3^-^^^^^^^ common denominator-for ?? /=lX6\ ^^^ 4 •__ 4 ^T will then be found equal ^^ ^ ^^ '^' '''' ^ (""15X5) denornl7or'Srnu£ber'o?f-*'P"^^ ^^ not contained in the the method given in th?rf/;!- """' expressed by an integer , The rule wiirevUntly anSv""?/'.^'- T"^ ap^plicable ^''' Pl>ed by a fraction-s"fce^£ Ir^^ '""^T^ '' *« ^e multi- whatever order the factorrart'tn? rr«L'1"'^^i^ '^''^'^ i» VULGAR FRACTIONS. 153 38. Tho integral quantity wlilch is to form oiio of the factors may consist of more than ono deuumiuatiou ExAMi'LE.— What is tho f of £5 2s. \)d. i £ 8. d. k 5. d. £ s. d. 5 2 Ovj ^5 2 9x2 _3 g 0. 1. fX2=l|. 2. 5x8=6^. 3. f,Xl2=:10J. 4. Jxl2=91. 5. VVx30=14. 11. i2x«G=34. 12. i«x20=l9. 13. 22x|=4f. 14. AXI7=U 15. l43xH6i|. EXERCIBKS. 6. 27x1=12. 7. Axl8=3«. 8. 1|X8=71. 9. 21xiJ=9. 10. 15x1=3. 16. Plow much is -^^ of 26 acres 2 roods .? Ans 20 acres 3 roods, 17. How much is \^ of 24 hours 30 minutes } Ans 7 hours. 18. How much is /jVa of 19 cwt., 3 qrs., 7 & .? ul7w 7 cwt.,3qrs.,2 1b. 19. How much is if of dC29 } Am. £\y ==£Q 195 39. To multiply one fraction by another — Rule. — Multiply the numerators together, and under their product place the product of the denominators. Example.— Multiply | by |. 4 5^4x5 20 9 6 9x6'^54- 40. Reason of the Rule.— If, in the example <,nv..n, wo were to multiply f by 5, the product (^^O would be tinea too great— since it was by the siMh part of 5 (^), wo should have multiplied. -But the produ.;t will become wliat it ought to be (that IS, G times smaller), if we multiply its denominator by b, and thus cause the size of the parts to become 6 times less. _ We have already illustrated this subject when explaiu- mg the nature of a compound fraction [11]. 20. 21. 22. 23. ■LvS 3 5 l^Xt!" ^ ^ l^A. — -'8' I v^ .1 x^= EXERCISES. XV4 4 8 48 1 24. 1^X4! 25.JXfXfV=^5 ffS- 27 3 14 vl?7. 3^3 si) ?.4 28. 29. 30. 31. 32. How much is the %■ of 3. > y} ,,? i X V 1 1 fjXA=fV I 2 -^ 8 — 1 6* -fins. -i. 33. H ^w much is the f of f ? A //y t ; ' ■f^"' IM Vl/LOAR FRACTIONa. of a fraotinn W "' "• . ^e8»"«8, the multiphcat oa that of division ; and the number said in hi . u-v^i must bo made loss than boforr ^' multiplied mkfd n^u"^^^ ' *'^^^'^"' ^^ ^ ^'^ ---^er by a raf/tn'T;?^*^"'^ '"''','^ °""^^«^« to improper fractions r24J, and then proceed according to the last rule Example 1.— Multiply J by 4|. "*!==*? J therefore ^ x 44=A y * > ~ 1 2 3 Example 2.— Multiply 5J bv 63 *'^" 52=V. and Gf=3^- tterefore 52x6f=:V x^^=^^^ „. EXERCISES. 37, Ax8Iy» v.. '■=;,.. £' Pixl3|x6|= SoTiJ. ^ 44. m. is t.o p.oa„et of e:'^1tt77> ^45.^ What i. tho product of f of f , and | of 3f » 8 4' # r f . ULOAR FRACnONS. 105 44. If wo perceive the numerator of one fniction to bo tho same as the denominator of the other, wo may, to perform the multiplication, omit the number which m commou. Thus f X5 = f. «.;'!" 51* '* ?f, *""'® f 'livJding belh tl.e numerator ami fbM.o- mmator of the product by the saiuo nuiubuv— uud theicfore does not alter its value; since lut^itioia o^y= 5 -ti~i}' 4.). Somctnnes, before performinju; the mnltiplic.-.tion, we can reduce the numerator of one fnicti.>n nnd I ho denominator of another to lower terms, by divi.Iin ' both by the Ranio number :— -thus, to multiply i by •» ^ Dividing both r; and 4, by 4, we get in their placos, d and 1 ; and the fractions then are A and J which multiplied together, become -^-X 4 = 7'-. " ' tnJni-^/" *^'^'''''P« as dividing the nuincrutor and denomina- tor of the product by the sanio number ; for 8^7-bx7-j-4-2x'7 V=2^7/ ^H' QUESTIONS. 1. How isafrnction multiplied by a whole number or the contrary ? [36] . 2. Is it necessary that the intccror which constitnfos one of the factors should consist of a single denomina- tion.? [38]. ° 3. What is the rule for multiplying one fraction by another ? [39] . "^ 4. Explain how it is that the product of two proper tractions IS less than either .? [41]. 5. What is the rule for multiplying a fraction or a mixed number by a mixed number ? [42]. 6. How may fractions sometimes be reduced, before they are multiplied ? [44 and 45] . 166 VULGAR FR ACTION'S. WVlSIOxY. tl.o whole 3^ and tui;';"'","' "'" «■''*""" h inerator. ' I^"' ""= Product under it, „u. Example. — ?--i-4 ^ ^ 47. ReASOJV of THF TJirrr. Ti t ., for instance, is to make ifSr '''"^''^^^ '"^ quantity hv 8 ^t is evi.leut that if, wwl L i lo .?'^''"'^' *''"'^" ''^'^'"ro- J3 t same, we make theiTlhA If^'' \^'^ '"'"'^'''' ^'^ *''« P'lrts the Jteelf 3 times Jes.s- !Urct to muU! f ''..^^ T^^^^^ theSvact a is to divide the fraction bv tL i-,, ^'"^ * '^ denominator by 3 A similar cifect wT k^ f """; ^^umber. '-^^ ^' ^iule we leave the .^. of^Sp^t" ;^ ^ "^^^f!. --|"-| if. --^.. 3 times less; thus '--^iJ^/J "^ "'^'^^ ^^-- numerator is not nh„ '^ ", '^ ~>' "^'"^ ^"'^° *^'« of -^ comi.l,i lZfoT[l'2] " '^'^ ''^P'""'«i ">o nature 1. II_i_9 4 2. |4^^"7 4- ^-9=^. ESERCISK3. «• H-8=,, 12. -^,-M4_/''^^ 7. A->14= • . wlieu we multiply rdtido?s'° ^'^ ? "^^^^or, th.-,,f, nator-bj the sa no r mm or w ?""^*^^'^l«^ ^"d denonu-' «nce we then a tlTZl r ""^ ^^^'^■^" ^^^^ ^^^1"«- dcoroase it. ' '" '"'"" *^^^> ^^I^^^^J increase and 4f'- To divide a fraction hy a fraction- llur,E.— Invert the dlvf^ri /"''^'^^^n— '^o^ed), and then placed ^i/tL7^r ^' '^ ^^ ^^■■ multiplied. ^ " ^"^ fractious were to bo VULGAR FRACTIONS. 157 Example. — Divide | by f . 5_^3_5 4_5x4_20 7*4 7^3 7x3~2r Reason of the Rule. —If, for instance, in the just given, we divide | by 3 (the numerator of the we use a quantity 4 times too great, since it is not I the fourth part of 3 (|) we are to divide, and the (£y) is 4 times too small. — It is, however, made what to be, if we multiply its numerator by 4 — when it 1^, which was the result obtained by the rule. 50. Tho division of one fraction by another may bo illustrated as follows — example divisor), y 3, but quotient it ought becomes 5 . 3 '.A «n •a * ns - — The quotient of f-r-f must be some quantity, whic-b, taken three-fourth times (that is, multiplied by -^-), will be equal to f of unity. For since the quotient multiplied by the divisor ought to be equal to the dividend [Sec , II. 79] , -f is f of the quotient. Hence, if we divide tho five-sevenths of unity into three equal parts, each of these will be owe-fourth of the quotient — that is, precisely what the dividend wants to make it four-fourths of the quotient, or the quotient itself. 51. When we divide one proper fraction by another, the quotient is greater than the dividend. Nevertheless such division is a species of subtraction. For the quo- tient expresses how often the divisor can be taken from the dividend; but were the fraction to be divided by unity, the dividend itself would express how often the divisor could be taken from it ; when, therefore, tho divisor is less than unity, the number of times it can bo taken from tho dividend must be expressed by a quantity greaUr than the dividend [Sec. II. 78] . Besides, divid- ing one fraction by another supposes tlie multiplication of the dividend by one number and the division of it by another — hut when the multiplication is by .a ffrea';er 4 m 'hfl fW •4 2 158 VULOAU FRACTIONS. must beicrea™!, " ''""""'^ ""^ '» •"> -iviJod 13. ^-^3=I^! 14. 4-^2=1 15. 14-1=13. EXEnClHKS. ?7- l"^H'- I i»--fi-^=i^' 18. j#-i=ii. 21. ?j.jrr 62. To divide a whole number by a fVaotion' minator of the product '" ''' '''''''''''''' '^^^ ^eno- Example.— Divide 5 by ^ 5^3^5x7^a5 ' 7 o "-" "5" • This rule is a consequence of Hm i..of . ^ t-cr may be considered a" "fr- t\ V •'' '^"'-^ ^'"^^^ """i" iDinator [14]; hence sl" J'^n^'x-- "'^ """''^^ ^'"' '^'""- b^'<S; dJn^SS U!'' '^''^'^ --'^-' «'-11 consist of Example. -Divide 17.s'. ShJ. }>y 3 22. 3-4-1=6?. 23. ll^/*.=,i.'u 24. 42-;;j,=gG4. I-\')SRCISES. 25. 5-i-i|=,')i 31. Dmde £7 IGs. 2d. by a yi., o,. , / ^ , 7 32. Divide ^8 13.. 4d by r J ' f n J"^^'^ 3.Prv^e^5 0..1..by^r j^;; ^^^ ^ o^:S. 10 divide a mixed nmnbnr h,r o T.ri i , or a fraction— '^•^ ^ ^'^^^^^ number and 49]. ■'' ^''''^'' '' ^^^"^^^ directed [40 Example 1.— Divide 9f by 3. 9?-^3=9-f-3-f■;^ -^3=34-1=3' Example 2.— Divide 14^3- by 7 ^' ''' '.4''='*^'' therolore lM--^2=',F-M = '^X^='?-« '■■.■^n:'^:-'^0^m^f ''■:■''' .5 .'J VriLrjAR FU ACTIONS. 159 54 llEx^oN OF THK RuLE,~lji the first cxnniplc we have divided each part of the dividend by the divisor auu added the vesults-which [Sec. II. 77] h the same as dividing the ^•liole dividend by the divisor. In the second example we have put the mixed number into a more convenient form, without altering its value EXERCrSES. 34. 8|^17=f|. 35. 51|-^.3=lK\ '151 rs- .1^ fl450 ■''Tf. 39 433S_:_41 , 40.- iifi^MSffir 41, 18-8X^^5— 19*773 42 loVJ^irliim, 43. 18±-Ail=ii«7 •'"^ '■v^' 55. To divide an integer by a mixed number Rule.— Reduce the mixed number to an improper fraction [24] ; and then proceed as already directed 1 52]. Example. — Divide 8 by 4§. 41=%^ therefore 8-f.4f=8-^ 2^3=8 x./j^Uf. Reason OF THE RuLE.-It is evident that the" improper fraction which is equal to the divisor, is contained in the divi- ilend the same number of times as the divisor itself. ' 1 44. 45. 4S. 49. 46 14-^l|=7^-V. 47 21^ll3*-=ii-i . 3 5^- n^d. EXERCISES, 5-f.3l=lf, 16-i- 1112—113 3 Divide £7 16s. Id. by 3i. Ans. £2 6s. Divide £3 3s. 3d. by 4i, Ans. Us. O^d. 56. To divide a fraction, or a mixed number, by a mixed number — "^ ron'''''''75'''^''''' "''''',^ numbers to improper fractions L-J4J ; and then proceed as already directed [49]. Example 1.— Divide | by 5 J. 6l=.f, therefore | ^5^=^^5/X,3><^^^,^^ Example 2.— Divide 8j9j- by 7|. 8A=fJ-, and 7|=V, therefore 8X-^7#r^"_..y-r.7 w 47 Reason of the Rule. — We (n.<i in tim loof •, i \ , ehango tho K,ixed nvmbcr, into *o h e" fm J ' cot S'j dmcled-without, however, altering their value °'"'"'"'"'"'' o 11 fl 56 2 '1 i ^ij 160 VCLGAR FKACT/ONS. 50. ~'--^.TJ. — •lii 51. ^14-41=11* 52. J<.^3A=Vi3 53. U^jlll^¥^^- 54. MJ-sCfT 2 • ^3 ^32. EXERCISES. 55. 82 rV— 26 /'-=<= '?«■■■ 50 ioM\i-^*'',ti<'>^^- 57. ^^:^8^=::i^3»^^^"'"--- 59. 2|J.34+i^e-..h, 58 When the divisor, dividend' o/hnfJ. " poand, or complex fraetimis- ^ ^'^^'' ^'"'^ "^""- Wrsr^'p^Xm^'ft^ -del,, to sin,.e which are cLCuTL^d tt 7^ ^« ^^'"'^^^ are complex ; tLntr'ocTe'd ^^l^^S^Z^^:^) ^Example l._Divide 4 of | by f . '" ' ^"''^ f of «=3o {-39j^ ^j^^^^^^^^^ fX«lo=-^n_no^._,,, Example 2.— Divide ~ bv •' i=4% M, therefore |-g=^^.=^x|=^, o rn ^ o EXERCISES. 01. 4fi-l— S^VJ* «;n43 62. '8~ IT * 21 2 •> 63. H^sy_7_n7 97 • 3'^i5— im- o4. . — 1- S.~r-on UO. — _i;.2'v « OfTt 19 '" <i 60. ~_i.2v5 Q231 7 QUESTIONS. I. How is a fraction dived bv an integer ? r4fil o. J^ixplam how it ocputq f^.;* *i """.• L'*-*J. fractionsis sometimes greZthal t fT' ?i '^" 4. How ;.! -, »i,,i 8'"'"" man the divider,-. J fsi i [62]. ' " "'""^ """'''«'• divided by a finetlW? ggj' number, by a mi^d number ? [55 and to.I- r'l;tVot oSpTe;t^^^^^^^^^^^ [fir. " .*^Jr:.. .'■^ -„ :t^ .= VULGAR FKACTIONS. 16J MISCELLANEOUS EXERCISES IN VULGAR FRACTIONS. acres, 3 roods ?* Ans. 1. How much is ^ of 1S6 20 acres, 3 roods. 2. How much is ^ of 15 hours, 45 minutes ? 7 hours. 3. How much ir, //f% of 19 cwt., 3 qrs., 7 Bb .? ^7w. 7 cwt., 3 '^rs., 2 tb. 4. How much is ^V/^ of £100 ? Ans. £3Q 95. 5. If one fanii contains 20 acres, 3 roods, and another 26 acres, 2 roods, what fraction of the former is the latter > Ans. -^^\. 6. "What is the simplest form of a fraction express- ing the comparative magnitude of two vessels — the one containing 4 tuns, 3 hhds., and the other 5 tuns, 2 lihds. ? Ans ' " JlS. o n 1. What is the sum of | of a pound, and f of a shilling ? Ans. 13^. lOfri. 8. What is the sum of |-5. and ^ul. ? 9. What is the sum of dCi-, a/ 36'. If-irZ. 10. Suppose I have -^ of a ship, and that I buy y\ Ans, 7j\d. and y'jf/. } Ans more ; what is my entire share ? Ans. J-i. 11. A boy divided his marbles in the following manner : he gave to A ^ of thorn, to li j\, to C i, and to D i, keeping the rest to himself; how much did he give away, <and how much did ho keep ? Am. He gave away tVo of them, and kept j\?^. 12. What is the sum of | of a yard, j of a foot, an(J 4 of an imh .' Aiis. 7 inches. ■'3. Wha. is the difference between the | of a pound and o}d. ? Ans. lis. 6^d. 14. If an acre of potatoes yield about 82 barrels of 20 stone each, and an acre of wheat 4 quarters of 460 lb — but the wheat gives three times as much nourish- ment as the potatoes ; what will express the subsistence given by each, in terms of the other ? Av:<. The pota- toes will give 41 } times as much as tlie wliL^a, :, and tho wheat the ,£j\ part of wliat is given by the poiu-toes. 15. In Fahrenheit's thoniiDnioter "there arc 'SO de- grees between tho boiUng and freezing points , in that 163 iniClMAL rUACTIOiVS. of Rcaumav only 80 ; what fraction of a do.rcc in tho lat or oppresses a cjegrcc of the K,r,ner r X? ' " *^'' isak.'nt ia -^T^' ^'^ 'f '''''' ^" t^^« United" Kingdom w about 34 inches jn aepth durino- tlio voir in fl,o t ,• of hi boii /!'" """"":>"'''. "■' 22,480 ; what fnlctiou 1. «i^ "* ^ '"■S"""! ''^l"™^"^ tl'^'-t of Chimbora.o ? jjoi, w^i r » r zr To=x™rri fraction of the latter ? ^w^. f.i. "^ '-^prc.ssca as a DECIMAL FlliVCTlONS. 59. A deun.:il fraction, as ah-eady rcmarlied Tl-^l Sinoo the division of\Su™l;Cof\t ci:!! tc'l™; by .ts d,3n„™.„to,-fro,„ the ve..y natu,ror ,!^ ^ S infnf fl J""'^ Pra-formod by moving the do.nmil 60 It is as inaccurate to confound a decimal fraction with the corresponding decimal, as to confoZd a 4w fraction with its quotient.— For if 75 i< th. fi , from eit,-:^"""™* °' ^ "'• "' ^^- "'l Ciually'distior; mnl 'f. ^."<"^™»I « cliangcd into its correspoudi,,.. deci- ml fraction by p„tting unity witl. as manv cypfcs 4 f»"^ .';'''"" 'r^' "■'' '!=«'"■■>■: point. Thus -Oe^fi^ I O 1) ) '^^ ,•5 (1 4 r, I J DECIMAL FnACTI0N3. 163 gi'Go in tlio I Kingdoiu the plains ; mt fractioa Iiigh, and at fractiou limborazo ? fissure or bet as tlie lountairis ; sscd as a :ed [13], lilt hand. fra<.ition. I fraction nutation deoiuial ho equi- facility, ty by a nt three fraction a vulgar q.wtknt ; so also distinct ng deci- )hers as nator — • 5646= ^2. Decimal fractions follow esactif UiC same rules as vulgar fractions.— It is, however,* generally nioro convenient to obtain their quotients [oG]^ and then per- form on them the required processes of addition, &c., by the methods already described [Sec. II. 11, &c.] 63. To reduce a vulgar fraction to a decimal, or to a dediiial fr actio, i — IluLE. — Divide the numerator by the denominator- tins will give the required decimal ; the latter may be changed into its corresponding decimal fraction— as already iokicribed [61]. ExAMPLK 1.— Reduce I to a decimal fraction. 4)3 Example 2.— What decimal of a pound is lid. * 7^/.= [17] £i^- but £5Vo=^C-0032, &c. This rule requires no explanation. EXERCISES. 1 7 _8 7_5 3. ^V— 36. 4 i' — jK'L ^' 4 100' 5. •! 625. 6. -^^=-973&o. 7. J=-5. 9. -j«,/.,=-90476, &c. 10. |=.8. 11. /^=5625. 13. Ileducc \2s. Gd. to the decimal of a pound. Ans 625. 14. Reduce Ids. to tlie decimal of a pound. Ans. -75 15. lleduce 3 quarters, 2 nails, to the decimal of a yard. Alls. '875. 16. Ileduce 3 cwt., 1 qr., 7 lbs, to the decimal of a ton. Ans. -165625 64. To reduce a decimal to a lower denomination . IluLE. — Ileducc it by the rule already given [Sec. III. 3] for the reduction of integers. I^lsAMPj.K 1.— Exjjress £-6237 in terms of a shilling •6237 20 . li ft 'in 11 Answer, i2'4740 shillings=.CG237 B! 'if HI-. 164 DECIMALS. Example 2.-Rcducc i;.9734 to shilling. &c •9734 20 19-4G8() 8hIllings=X-9734. 5-G160 ponce=-4C85. 4 2-4040 favthinffs=-G10^Z Answer, X-9734=las. 5 J,/ ro^S-oI'^^^^^^^^^^^^ a, wc. given of » shilling by rrroJuccs rt ■^'"'''l,''^ "8,'l'° <1««"""1 J'Wmy. MultiDlvinff t^ryj!, ? '^.t"™ ""'' "'" ■)«"»"'>' of a EXERCISEa 23. WJuat is tlie value of £-80875 ? yl,,,. 17,. 4^,1 -4. What IS the value of £-d375 ? 20. I ow much is -875 of a yard. ^ ^M,s^L^^^^^^^^ 27. AVhat is the value of £-05 ? Ans. Is ^^f^- How much is -9375 of a cwt. ? Ans. 3 qrs., 29. What is the value of £-95 } An^. \Qs. 30. How much is -95 of an oz. Troy ? Ans 19 dwt 31. How much is -875 of a gallon > Ans 7 nints 28'%'':^.r''' ''-'''' ^''^ ''^y^ ^1-// hours, ;J^' !^!'^ f^"owing will bo found useful, and—befnr, n nnatcly connected with the doctrine of SctionT^ may be advantageously introduced here : '^''^^^^"^-- io imd at once what decimal of a pound is enn.V. Whor;l"""'^" «f ^^^^. ponce, i?r '^""" \V on there is an even number of shillings- ' a pound" "''^'' '^''''' '' ^' ^'''^^ ''' ^^^y t^^^tbs of DECIMALS. 165 ivcre given i it to shil- l»o decimal cinial of a I'educcs it 175. 41(1 ■ , 2 nails. • 3 cwt., 3 qrs., 19 dwt. pints, hours, ins. 1 5 — being tions — • c-qniva- ?tlis of E.VAMrLE. — lC)S.=£-8. Every two shillings areciiual to one -tenth of a pound; there* fore 8 times 2s. are equal to 8 teuths. 67. Wbon tlio nunil)er of shillings is odd — lIuLK. — Considor half tho next lower oven number, as so many tenths of a pound, and with these set down 5 hundredths. Example. — 15.s'.=£-75. For, 15.? — 14S.+1.S. ; but by the last rule 14s.=cC-7 ; and Binco 2s.=l tenth— or, as ia evident, 10 hundredths of a pound — l.s.=5 humlredths. 68. When there are pence and farthings— K,uLE. — If, when reduced to fjirthings, they exceed 24, add 1 to the number, and put the sum in tho second and third decimal places. After taking 25 from the number of farthings, divide the remainder by 3, and put the nearest quantity to the true quotient, in the fourth dechnal place. If, when reduced to farthings, thoy are less than 25, set down the number in the third, or in the second and third decimal places ; and put what is nearest to one- third of them in the fourth. Example 1. — What decimal of a pound is equal to 8,J(L T 8J=35 farthings. Since 35 contains 25, wo add one tc ■ the number of farthings, which makes it 30— we put 36 in the second and third decimal places. The number nearest to the third of 10 (35-25 farthings) is 3— we pvit 3 in the fourth decimal place. Therefore, 82=£03G3. Example 2. — What decimal of a pound is equal to 1:^(?. 1 1^=7 farthings ; and the nearest number to the thir*! of t is 2. Therefore l|tZ.=£0072. Example 3. — What decimal of a pound is equal to 51(1. "? 5](/.=21 farthings; and the third of 21 is 7. Therefore - 3](i.=i:0217. 69 Rkason of the RuL,E.--We consider 10 farthings as the one hundredth, and one farthing as the one thousandth of a pound — because a pound consists of nearly one thousand farthings. This, however, in 1000 farthings (takeu as so many thousaudtlis of a pound) leads to a mistake of about 40 — Binco ,fil=(not 1000, but) 1000—40 farthings. Hence, to a tliousaud favihings (considcveil as thousandtliB o^ a pound), ii / /THs: .\i * '4 1C6 11 at' CIRCULiTlJtG UE^.jiAl.,. correction "liouhhtill bo I,, , M ,";;•■ "• " "'''"'" or .«.«w one, i„ t,,o/„°!;?raii;f„,'fr;s;'' ''* KXERCISEfl 18 7 i ft. ==:£ -0822. J9. ^-'7 5*. I0rf.=je27-2916 tljo •ne tiiat, ua must be nunibor, that the number, 20. 4,,.3|,/.=£.7i55 ^-i. £42 ll5.6R=i;.42677 ^-, in an/dtilrora'X^i:::!^"- "^ *"'»o-. ponce, tJonsMer the digit in ttc™?! ?,,'«" '',""'' °™- 6, If It 18 not loss thau 5^ r,„„i /f''''' ^n'>'™o'ing ;«.unito of farthings ; and s, b ta 1 "" •, f' '" ""^ "'"-^ >f It exceeds 25. *"'" "'"'^ f™ni the result ExAMrLi:.-je-6874=I3s 9rf 4r^^^t^::^^^:^i^^ -«- widths tiie remainder (reduced fn f i, ^^"n^/redtJis and adding sandths, we have^T thousandtt r^^^^^^l^- *^ *he tS exceed 25, we subtract Svtfe i™ ^vhich-since they of farthings. ^-6874 theJefoS i T' ?^/^ *^« ""™be? tarthmgs-or 136-. 9d. '"^^®^"^®' i« ^qual to 13^. and 3G "- -^^ ^^"owa fro. t.e I.t three^being the reverse of |;:|i CmzVLATim DECIMALS ''w»yi;ra"f 'c "ottro?^ ?^*'?'' [See. H. 72], number by a„otlor?-ll ST' ".''° '""''^ ™« 'ta..Iy recur, 0^0^,^ a'^^-^E' T *'^'"'; ™"- CIRCIJI,ATI.V(; BE(,...IAL8. Ul docimal is i)roclue( il.— Tho decimal ia euiJ to bo termi- nal, ,'.' there is an exact i|uouciit — or oue which loaves no romaiiider. 72. An iutormiiiato dcciiiKil, in which only a ingle figure is repeated, is callc ' a rppettnd; if two or more ^ligits constantly rccnr, thoy tonn a. periodiai/ '--' \, Tiins ;77, &c., is a repcfnnd ; hut -59759/, ..o. i.s a periodical.^ Vov the siko ot brevity, tlie repeated digit, or period is set down but once, and may he marked as follows, -5' ( = '555, &c.) or -M'jri' (-- 193493493, &.c.) Tlio ordinary method of marking i\n: period is sonie- wliat different — what is liore given, howcv<T, seems preferable, and can scarcely be mistaken, even by those in tlio habit of ^ -^ing the other. When the d imal contains only an vnjivile. M-t — that is, only tlu; repeated digit, or pei-iod — it is u pure repctcnd, or a p?/. rc_ periodical. ' But when there is hoth a finite and an infinite part, it i.-s a mixed repetend or viixe.d circulate. Thus j*'^ (=V)oo, &c.) is a pure rcpotcnd. '578' (=-57iS88, &c.) is a niixod ropotond. '397' (= o97397;">97, ka.) is a pure circulate. 8G5^G427r(='8G5G427164271G4271,&c)is a mixed circulate 73. The number of digits in a period must always ba less than the divisor. For, different digits in the perioci suppose different remainders during the division ; but the number of remaind;MS can never exceed — nor even be equal to the divisor. Thus, let the latter be seven : the only remainders possible are 1, 2, 3, 4, 5, and 6 ; any other than one of these would contain the divisor at least once — which would indicate [Sec. 11. 71] that the quotient figure is not sufficiently large. 74. It is sometimes useful to change a decimal into its equivalent vulgar fraction — as, for'instancc, when in adding, &c., those which circuhite, we desire to obtain an exact result. For this purpose — IluLE — T. If the decimal is a pure rcpclevd, put the repeated digit for numerator, and 9 for dcnominato.-. II. If it is a 2>ii''& puriodimly put the period for numerator, and so many nines as there are di<;its in the period, tor denominator. H IMAGE EVALUATION TEST TARGET (MT-3) i: ■V'J- •ip M^ ^ A' W- W^r / ^ 1.0 1^ 1^ m \m 142 1^ 136 22 \L. liiH I.I 1^ RRffil 1.25 12.0 1.8 14. ill 1.6 V] <^ % /a /a w om 9 V// Phntnoranhir Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 V iV ^ ^^ \ \ A 6^ >^ m '' ' - ,""■ > »'* S"y ■ . „"-;S,' >. iC'"" IG8 CIRCULATING DECIMALS. ^^ Example l.-What vulgar fraction is equivalent to -2' 1 Example 2. — What *7S54'? Am. 7S54 VW99- vulgar fraction is equivalent to we multiply two equal o7anHL?'""i?^^^^' '^*'-) ^^r i# quantities'ihep?oclltwiarbceoVar "^"^' °^ '^ ^^^^ Fo?1:E ^i;;-^ ^£^X^^T'' -'']' - quotient. OS 100 kundredhi3'\Z}ihTV^.^i'c,'^'^''^^ ^e considered 100 ten #/.«W "jM. Sefore t wi ^- "^^ ^' °^^^'« will be one /.n S.aS/L ^ Z.^? ' ".T^ "'^ '^'^«"^»<^ remainder, must, in the /amP L ^ k ' ^"P *^" thousandth, the eth.s; and the nexrquotiTt wIlL "'^^^'f P^^d as ten milHon. and so on wi irtl?e oS ni r .""^ '"^'^''''*^'^''«^ -OOOOOI-- will be -Ol-i^mi^Som^^^ together, by -^01'. ^ ^^wi^^vc, or 010101, &c.--representod ■ -2^5- (==37XB'fl=37X.^0r) will e-ive -3-37^7 Xr^ quotient. Thus ^■o/o/, &c.— or 010101, &c 37 a 3 70707 30303 Tr, +1 .. S73737, &c.=37v-m' digits as a period, will bl eauaVto a v ,l ? ^?-"°« ""^ '« 603 3003003003 600600(5006 5005005005 T« *!,« ^6^5^3503503, &c=5fi3v\nni b >e a cucuiating decimal having these CIRCLI.ATING DECIMALS. 169 J'pita as a perio(J. — And, consequently, a circulating decimal linving any three digits as period will be equal to a vulgar truction having the same digits for numerator, and 3 nines lor denominator. We might, in a similar way, show that any number of digits divided by an equal number of nines must give a circulate, «ach period of which would consist of those digits. — And, Consequently, a circulate whose periods would consist of any digits must be equal to a vulgar fraction having one of : s j'.eriods for numerator, and a number of nines equal to the number of digits in the period, for denominator. 76. IF tlic decimal is a mixed repetond or a mixed nrculate — lluLE. — Subtract the finite part from the whole, and set down the difference for numerator ; put for deno- minator 80 many cyphers as there are digits in the Jinife part, and to the left of the cyphers so many nines as tliere are digits in the iiijiniie. part. ExAfdPLE. — What is \i\Q vulgar fraction equivalent to •97^8734' ? There are 2 digits in 97, the finite part, and 4 in 8734, the intiaite part. Therefore 978784-97 978G37 . ,^ • , , .... ~mm~=WMO' '' *^^ '''^1'"''''^ ^^'^Sar fraction. 77. Reason of the Rule. — If, for example, we multijdy •97^8734' by 100, the product is 97 •8734=97 4--8734. Tliis (by the last rule) is equal to 97-|-|o|^' which (us Ave multiplied by TOO) is one hundred times greater than the original quantity — but if we divide it by 100 we obtain TVo+iT-sMfis-'fn. "wlucli is equal tjio original quantity. To perform the addition of y^,- ^^^'^ irf sffiW' ^e must [19 and 22] reduce them to a common denomin?».tor — when they become 97X991*^00, 878400 97X9999, 8734 99990000 ' 99990000"" 999900 97x10000-1 , 8734 10000-1) 970000—97 ' 999900 97x10000-97 8784 "999900"^ 999900 ^999900 ~ 999900 8734 978734-97 978637 . 999900 "^999900~ 899900 ""999900' "^^^" ^^ exactly the result obtained by the rule. The same reasoning would hold with any other example. EXERCISES. 7. 1. 2. 3. 4. 5. 6. •^8'=5L • 73'=T^- _14 5 ^057'=rg?X ^145'= ^057'= 8. 9. 10. 11. 12. ./)74' 574 uu -.u — onno- •147^658'=:iil^ii 875-49vG5'=875*^^V' 301-8275G'=:301^;^^a. 170 ^•^KCULATINO DECIMAL,. vaient vulgar fractio, '^d "° *''"'"''^ "> "««• o„"i. ''«'«, &0, liie other <le",l!" '""■^ ?''''' -"id subtract put Joivn so jaanv of *i "'"'«'/ takin- ca.rtl acourac,. """^ »' 'l-eu, as will ^secure "suffieio,;: dered as :^2i£2«;s\ .„ *• ^tus a (const 5 >'«'^^g-™o.act,„otiont;soalso W'U i (considered as ™i£2*?x , 2 ) i^ni \ will not siyo »»; fe4 (considered as'-^MSSl^, „, IMJmndn^itl.s ^ "■■.•'^nkr reason 4 «/,„ '-"i->oo4 (considered asl°i-i!s;. 400hXdS: %l ''rj„1(';' ^■''.'"Ij contain 7^ ^^ '' »« will be eqtl to™e Zlt™ '""^y depi,nal places eontamed as factor i„^S 1' "™?''°'- "f twos, or fives therefore i2j£f^ " ^^ (^XS; ; and '-^^ place 'see. ^7^1^^""" ''^' ^"^ ''' quoaeoi. ^^- ^^J), that n, ,,,, ^.^.^^j ^^ > 2X2/ ^^" gJve two decimal places • ho. CIRCULATING DECIMALS. 171 '■gJ, it is not tlieir ofjui- nd subtract "ig cava to ■e sufficient ?ar fraction ' its Jowost ctors (fac- ' ictors, can or noitlior tliese — as oir inulti- ictly con- I (eonsi- * ', so also ' noi givo idredths. ct quo ii'Gdtlis, places 1* fives, riginal 'found ; and in the al as 5auso ■itor, •s so 30 tenths niany tenths ; for — ^ (=4 ) cannot give an exact quotient — 30 being equal to 3X2X5, which contains 2, but not 2X2. It will, however, be sufficient to reduce , , , , 300 hundredths the numerator to hundredths : because -. ' 4 will give an exact quotient — for 300 is equal to 3 X 2 X 2X^X5, and consequently contains 2X2. But 300 hundredths divided by an integer will give hmulredths — or two decimals as quotient. Hence, when there are two twos found as factors in the denominator of the vulgar fraction, there are also lioo decimal places in the quotient. 4*V \r^ o \^ o sy o sy r j contains 2 repeated three times ^ ,«X'*X'^X<J as a factor, in its denominator, and will give three decimal places. For though ]0 tenths — and therefore 6X10 tenths — contains 5, one of the factors of 40, \\ does not contain 2X2X2, the othr;> ; consequentlj' it will not give an exact quotient. — Nor, for the same reason, will 6X100 hundredths. 6X1000 thousandths) 6 X 1000 thousandths will give one — that is, j^ (=4V) ^'"^ leave no remainder ; for 6 X 1000 (=6 X2X2X2X5X 5X5) contains 2X2X2X5. But 6X1 000 Ihausandths divided by an integer will give thousandlhs — or threa decimals as quotient. Hence, when there are three twos found as factors in the denominator of the vulgar frac- tion, there are also three decimal places in the quotient. 81. Were the Jives to constitute the larger number of factors — as, for instance, in /^ jf ^j-, &c., the same reason ing would show that the number of decimal places would be equal to the number of fives. It might also be proved, in the same way, that were the greatest number of twos or fives, in the denominator of the vulgar fraction, any otlber than one of those num- bers given above, there would be an eqaal number of decimal places in the quotient. 82. A pure circulate will have so many digits in its period as will be e(pial to the least number of nines, which would represent a quantity measured by the donoraina- 172 CIRCULATING DECiaiALS. bo equal to a fraction ),,„/„ J """ '"<"' " e'reulate will that IS, It will be eaual +n =. ^ • ^^^ ^^enomiuator— of which (the ierlToff}T' ^T'T^ '^'^ numerator ii^ the numemtor of the ^,-''''"^' '^ ^^^ ^' «^ ^'^^J quantity represen ^d VlL^ST^^r^^^^?^^^ ^ ^^^ l^r if a fraction having a^i^'nl '^- '*' ^^"^""nator. another which has a alpf f ^ t""'"""'"^^^ ^« ^''l^^l to of the latter is to the slm^'. '" ^'T''' ^^'' numerator the former-^in which ciTtho""?" '?'' '^^'" *^^^* ^^ nierator counteract ttrefe^^^^^^^^ ''^'^ the nu- denominator. Thus A-Vf , *^'^ increased sue of tlio «f H is 5 times Teaterd^ ' i'TT^ " ^^^' numoratur «nd -384615'=5 8 4 el?"! J ^V ^^"^e /^=.-v3846ir/ • and, thereforp Jk" V^ '^' ' A' '^^^°' ^^ ^^nai to If a# i5 . _ u, tuertiore, whatever mult nIp '^ft.iRi- • »»y*nf'" > - J:8 the same of 13 —But qq S • ^f^^'' '^ ^^ ^> ^^''^'^^-^ 13, consisting of nin^s T? . ' f /^^' ^'''' "^"^^^P^^^ -*' Then take f?r numel-ato" sn .1 ' "' T'? "'^^^ ^' ^^^««- lesser number of ^ n'; Vs ^^ , i^ '"'"^y' ^^ '^^ ^« that number of nines for its deomulltr'Tl.^'' that lesser this new fraction will fir^l ^'^l'^'' -^.ho numerator of equal to the origina fia^i^on "\ ^ ^''''?^ '^ ^ ^^''^''^^'^tc different from 3846 5 Tf '^- ^"^ ^^t^"^ ^«^ Period k circulate; there TrCefe'tr^r^^"' *^? ^'^"--' equal to r^^-that is two rii!' T t^'"'"* circulates for the same frac ioni-wh.V f "'''' ''^^''^'^ ^' --'otiont, is absurd to suppo eVa^^a^.v /' '""^'f^^'- ^ 'nee it multiple of 13. ^ "^ ^"^ ^"»^^«^ of nines is a rarTwh'a^ nSf 2^t1"'fo'"^ ^^^ ?^^^^- '^^ ^-te' of the vulgar fraction^-oduced tot ? '^! ^^^^^ominator For f7fil n «».•* ^^uucea to its lowest terms hand"i/'ti; 't??n' r\' "''• "^P''^- *»"- ■■%>" fi-tion,oI,tainXmL 'S,.e:?r,rt; "' f''" "''^'"• suppose the (ienominator of t„ ^^»' ';.VP'""-s would -tain two, 0,. fivci^i:^ t sniiSrt;:: CIRCULATING DKCIMALS. 173 to its Icwost 3irculate wiJl 01' its nume- iiomiuator — e numerator be as maivy tion, as the snouiiuator. is e(|ual to ■ numerator 'an that of of the nu. ' «ize of tlio numcratur momiaator 2. '3846 If/ ; 0.4. 0L5 . a u ? ft ft ) — ^, U9!i't>J.4 lultiple of 'r bo loss. 5, as that i«it les'ser ^orator of circulate period is culate of Le former •irculaios '"'otients ■incc it nes is a a finite >minator IS. he riojht vulgar 'S would tion to facturn could give cypliers in thnir multiple- of the vulgar fraction obtained from the If there is a finite the dimominat\7r ciiculate. 84. If there is a finite part in the decimal, it will contain as many digits as there are units in the greatest number of twos or fives found in the denominator of tho original vulgar fraction, reduced to its lowest term.s. ' Let the original fraction be -/-g. Since 56.=J2X2X 2X7, the equivalent fraction must have as many nines as ^vill just contain the 7 (cyphers would not muse a number of nines to be a multiple of 7), multiplied by as many tens as form a ])roduct which will just contain the twos a8 factors. But we have seen [80] that one ten (which adds one cypher to the nines) contains one two^ or five ; that the product of two tens (which add two cyphers to the nines), contains the product of two twos oi fives ; that the pror'.uct of three tens (which add three cyp-hers to tho nines), contains the product of three twos or fives, &c. That is, there will be so many cyphers in the denomi- nator as will bo equal to the greatest number of twos or fives, found among the factors in the denominator of tho original vulgar fraction. ]3ut as the digits of the finite part of the decimal add an equal number of cyphers to the denominator of the new vulgar fraction [7GJ, the cyphers in the denominator, on the other hand, evidently suppose an e(|ual number of places in the finite part of a circulate : — there will there- fore be in the finite part of a circulate so many digits as will be equal to the greatest number of twos or fives found among the factors in the denominator of a vulgar fraction containing, also, other factors than 2 or 5. 85. It follows from what has been said, that there is no number which is not exactly contained in some quantity expressed by one or more nines, or b;7 one or more nines followed by cyphers, or by unity followed by cyphers. Contractions in MtrLTiPLicATiON and Division (derived from the properties of fractions.) 86. To multiply any number by 5 — IIulb:. — Remove it one place to the left hand, and divide the result by 2 174 <'"\ri:.\'TroN'<<. KxAMrr.r;, . 7:!Ox/5~."^^h,^;..;^,, ^'- 1<» imiltipiv \,y 25 _ 3 divide liM^'""'"' "" 'i'"^»t^'y two places to tho left, and M\.\.AnM,K.— 0732x25.^ ''f.iaon..,, .«.,^n -^— , ; flH^rcfurc G/.!i»xi:5=G732x'"" *^- 'i'o multiply l,y 125 ._ " * di^:'^';::::;:i;'i ^^^^-^^^ ^j-- pi^^ec. to the loa. and liiOAsox. — 12-5— ''^»"- t) r Tr-^r^-^-^^- KVAMl'LE G85 XTSsTT*"' 5"" ^T .,n . - lOOx,/. * --I'JUX, : thei-fifore 085x75 = C85x 'H). To imiltiply by 35— left STi!;- l^^'t'S^tfo""?"!- ^"^ I'^-- *« tho I'luce to the left. ^ ' ^ *''' "Hiltiphcand removed one 'f~-fi'0. ' therefore G / 80Gx 35 =67806 x p!; J'' t'^' '7'^"^ <^'^'^'>^ the multipliers-- Fv^;7 v^ -^ '^^ '"' "^'"^'^^^"^ ^'^^tion. inverted ^L^.,...4.v.de 847 ,^ 5. 847..5=847-^ V^47x easy to divide, a. ro v^ultnAl'C"'^j'-,r^^". '^ ^'."^'^ ^^ liUAed number. " ^ "^ -^ •* r ^^> i^« crpnvalene DECIMALS. m ^=«^v"==:5'Vfto. tlio left, and o V ' " " QUESTIONS FOR THE PUIMf, 1 Show tliat a decimal fraclion, and the spond- ^ X ' the loft, and 'SG5='7o. "•f^ loft, then hy -1. 5. C75 = C85x «cns to tlio Jinovcd one == 109740Q = 67896 X ornsclves to ertod. y'=847x ■ide hy the 1 wJieiii we 3h'er v'ill is not so riuivalciit Iccmif *"6 "'-•'^"""'i «»>iu iiuu identical L^jyi. 2. How is a decimal chann;cd into a decimal frac iion? [GIJ. 3. Are the methods of adding, &c., vulgar and deci- *nal fractions different ? [62]. 4. How is a vulgar reduced to a decimal fraction ? [63]. 5. How is a decimal reduced to a lower denomina- tion .? [64]. 6. How are pounds, shillings, and pence changed, ai once^ into the corresponding decimal of a pound r [66, 67, and 68]. 7. How is the decimal of a pound changed, at once^ into shillings, pence, &c. } [70] . 8. What are terminate and circulating decimals } [71]. 9. What are a repctcnd and a perio Meal, a puro and a mixed circulate ? [72] . 10. Why cannot the number of digits in a neriod bo equal to the number of units contained in the divisor } [73]. 1 1 . How is a pure circulate or pure repetend changed into an equivalent vulgar fraction ? [74] . 12. How is a mixed repetend or mixed circulate reduced to an equivalent vulgar fraction r [76] . 13. What kind of vulgar fraction can produce no equivalent finite decimal ? [79] . 14. What number of decimal places must necessarily be found in a finite decimal .? [80] . 15. How many digits must be found in the periods of a pure circulate ; [82] . 16. When is no finite part found in a repetend, or circulate ? [S3] . 17. How many digits must be found in tho finite part of a mixed circulate t [84] . 18. On what principal can we use the properties of fractions as a means of abbreviating the processus of multiplication and division ? [86, &c.] til l<! 176 SECTIOJV V. PROPORTION. numbers are given a W]! ^ •T''"' '^ ^^' ^^^^^^^ tluco found. ^ "' * *^"^*^' ^'»«h ^s unknown, may bo shown by Hatton, in bis IrthJr i ?^«"''ate, as was hundred years ago '^^ '''^^" published nearly one p4oS,^::!l tXs:r^i^T;^ ^^^^ f « ^^ miportant prf clnles nnnn '1/ !? •?, , '^ ^''^^ ^^^^P^e but i he following tmth3 aro self-evident •_ quantity, 4 for instance Ix^ Id '• vm'^^ ."'"J'""" equal, we shall have 5X6+4=3X10+^ ' """^ ''™ equal ^':^l^^.:p:''^ij: ^ «"^" - ^. 11 the same, or erjinl r<»..^*-*- from others whieh a e Tual ^ ''" ''^ '''^''''''^' equal. Thus, if we subtCt'- f." ''"^'^f^^^ ^^^ bo quantities 7, a'nd S+Cweiu l'.? ""' '' ^'^ ^^^"^^ 7-3=5+2-3. And since 8=6+2, and 4=3 + 1. 8-4=0+2 -3+T PROPORTION. 177 tho golden ' it is termed . when thrco wn, may bo ' the simple^ divided into ate, as was 1 nearly one tlio rule of simple but •0 of ratios. 'i to 0(j[ual d tlio sum,. which aro kvhich are subtracted s will bo the equal same, or Thus if wo multiply the Oi\\\v^h TH-fl, and 10+1 by 3, bhall lia\ we G+(]V3=:U4rix3. And Kiucu 4 + 0— ID, and 3X().~18. 4+'JX3xG=i;5xl8. 0. If equal quantities aro divided by tlio same, or by • equtil (|uantitit'.s, the quotients will be oqunl. Thus if wo divide tlie C(iuals 8 and -1+4 by 2, we shall havo 8_4+4 2 T And since 20=17 + 3, and 10=:=2xr). 20_17+8 10"~~2x5" 7. Ratio is the relation which exists between two quantities, and is expressed liy two dots ( : ) placed be- tween them— thus o : 7 (reatl, 5 is to 7) ; which means that 5 has a certain relation to 7. The former quantity is called tlic onlccedevty and the latter the covscqi/rnl. S. If we invert the tci ms of a ratio, we sliall havo their inrenc ratio ; thus 7 : 5 is the inverse of 5 : 7. 9. The relation between two quantities may consist in one being greater or less than the otlier — then the ratio is termed arithmctlad ; or in one being some mut- tipk or part of the other — and then it is geometrical. If two quantities arc ecjual, the ratio between tlunu is said to bo that of equality ; if they are unequal it is a ratio of greater inequaUty when the antecedent is greater than the consequent, and of hsscr inequality when it is less. 10. As the cviithmetical ratio between two quantities is measured by their difference, so long as this difference is not altered, the ratio is unchanged. Thus the ratio of 7 : 5 is equal to that 15 : 13— for 2 is, in each case, the difference between the antecedent and consequent. Hence we may add the same quantity to both tiio antecedent and consequent of an arithmetical ratio, or may subtract it from them, without changing the ratio. Thus 7 : 5, 7 + 3 : 5+3, and 1 ->2 : 5-2, arc equal arithmetical ratios. Uut we cannot multiply or dih'ide the terms of an arith- I if It 173 PnoPORTlON. V r "■■0 c,,uul ; thus 10 : 5=12 : fi L ' •- ■ a ^u ","'" >-»t.o by the «u,uo number mthoutaltoring^I.e .aao U'us 7X2 : VX2=7 : 14-bocauso '^^^ 1 tion formerly mvcMi " Wl.^f f... 1- 1 f^J" q'les- 2U ?" «^l,{i • ' ,. ^*^'^'^ Inichon ot a pound is 7i* , — wJiich in rea ity moans " Wl...f .. / /• • there between 2\d. and a pound '' or ''WI ? '? '' consider 2U.. i w. nnn«\r 1' „„^/ ^V hat must wo or. If consider 9iZ It- ^^ 1"^""« 5" or " What mu: Tn fine '?wi;. -Ti '""f '^ ''^ P^^"^^^ «« "'"^y ;" » w ' ^* ^^ ^^>^ value of 2J- • 1" terms by the same numb rffif 1 9 • IZt t^""' "^ ratio a8jf^:«2 or n // ,m, ™ ** ;, ''™ '? 'ho samo f:4i^,iii,rjj:;°st'^^^^^ •tuuu nmtij rv!»sonta their ratio, aud unity. Thus PROPOUTION. 179 •n flio liLst example 9 : 000 and ,J^ : 1 arc equal ratios. H If, not iioc'jssary that wo hIiouKI bo able to oxpiess by int^igors, nor even by a finite decinuil, what part or mul- tiplo one of the terms is of the other ; for a geometrical ratio may be considered to exist between any two quan- tities Thus, if the ratio is 10 : 2, 5 ( V) is the quantity by which wo must multiply one term to make it equal to the other ; if 1 : 2, it is 05 (^), a fmite decimal : lut if 3 : 7, It IS M28571' (^), an infiuitc decimal— in which case wo obtani only an approximation to the value of the ratio. 13ut though the measure of the ratio is ex- pressed by an mjiiiite decimal, when there is no quantity which will exacfiy aerve as the multiplier, or divisor of ouo quantity so as to make it equal to the other—sinoo wo may obtain as near an npproxunation as we please^ there is no inconvenience in supposing that any one number is some part or multiple of any other ; tl'at is, that any number may bo expressed in terms of another— or may form one term of a geometrical ratio, unity being the other. "^ 14._ Proportion^ or analogy^ consists in the equality of ratios, and is indicated by putting =, or : :, between the equal ratios ; thus 5 : 7===Q : 1 1, or 5': 7 : : 9 : 11 (read, 5 IS to 7 as 9 : 11), means that the two ratios 5 : 7 and 9:11 arc eijual ; or that 5 bears the same relation to 7 that 9 does to 1 1 . Sometimes we express the equality of more than two ratios ; thus 4 : 8 : : G : 12 : : 18 : 36 (vcd, 4 is to 8, as 6 is to 12, as 18 is to 30), m'eana there is the same relation between 4 and 8, as between 6 and 12 ; and between 18 and 36, as between either 4 and 8, or 6 and 12~it follows that 4 : 8 : : 18 : 36— for two ratios which are equal to the same, arc equal t' each other. When the equal ratios- are arithmetical, the constitute an arithiMtiail proportion ; when geometri cal, a geometrical proportion 15 The quantities which form the proportion are called proportionah ; and a quantity that, along with three others, constitutes a proportion, is called a f&iLrth proportional to those others. In a proportion, the two outside terms are called the extremes^ and the two middle terms the means ; thus in 5 : 6 : :7 : S, 5 and 8 are tho m J- \ ■ A %• 180 PKOPORTION. extromos, 6 and 7 the meanf.. mien tlie same qiiantitr IS found m bolA means, it is called l/ie mean of the extremes ; thus, since 5 : 6 : : 6 : 7, 6 is tAc mean of 5 and 7. VV hen the proportion is arithmetical, t/ie mean of two quantities is called their arithmetical mean • -when the proportion is geometrical, it is termed their' ^e^Ts^- onml mean. Thus 7 is the arithmetical mean of 4 and 10; for, since 7-4=10-7, 4: 7: :7:10. i\nd8ia the geometrical mean of 2 and 32 : for, since 5 j 2 : 8: :8 : 32. > ^^^^ s— ai* 16. In an arithmetical proportion, " the sum of -the means is equal to the sum of the extremes." Thus, since 11:9:: 17 : }5 is an arithmetical proportion, 11-9^:=^ 17-10 ; but, adding 9 to both the equal quantities, we have 11 9 + 9=17-15 + 9 [3]; and, adding 15 To n'^'^To h^-' •^^-^+^+^'^=1^-1^+^+15 ; but H g^.9 + 1^ jge(^^j.j| ^^ ii + i5_sinco 9 to be sub- tracted and 9 to bo added =0 ; and 17-15 + 9 + i5-_ 17+9_since 15 to be subtracted and 15 to be added =0 • therefore 11 + T5 (the sum of the extremes) =17+9 (the sum of the mean ,.— The same thing mi^ht be proved from any other arithmetical proportion^ and, ■ therefore, it is true in every case. ■ l\ 17. This equation (as it is called), or the cqualitv which exists between the sum of the means and the sum of the extremes, is the te^t of an arithmetical proportion :— that IS, It shows us_ whether, or not, four given quantities corstituto an arithmetical propor-tion. It also enables us to hnd a fourth arithmetical proportional to three given numbers— since any mcc^n is evidently the difference between the sum of tlie extremes and the other mean • and any extreme,' the difference between the sum of the means and the other extreme A fiT '^r^ •-'^r\L? • 11,^^ *^' arithmetical proportioif,9 V ;,+- ^1^^ ' ^°^^' subtracting 4 from the equals, wo have 1 1 ,one of the extremes) =7+8-4 (the sum of the means, mmus the other extreme) ; and, subtracting 7 we have 4+11-7 (the sum of the extremes minus !no of the means) =8 (the other mean). V.'e might in the ^amu way nna the remaining extreme, or the remaining mean. Any othtr arithmetical proportion would hav6 PROPORTION. 18i answered just as well — hence what we have said is true in all cases. 18. Example. — Find a foiirfa proportional to 7, 8, 5. Making the required number one of the extremea, and Cutting the note of interrogation in the plac"e of it, we have : 8 : : 5 : '? ; then 7 : 8 : : 5 : 8-}-5-7 (the sum of the means minus the given extreme, =6) ] and the proportion com- pleted will be 7 : 8 :: 5 : 6. Making the required number one of the means, we shall have 7 : 8 : : '? : 5, then 7:8:: 7+5-8 (the sum of the extremes minus the given mean, =4) : 5 ; and the proportion completed will be 7 : 8 : : 4 : 5. As the sum ov the means will be found equal to the sum of tlio extremes, we have, in each case, completed the pro- portion. 19. The arithmetlad mean of two quantities is half t\\Q sum of tho extremes. ¥oy the sum of the means is equal to the sura of the extremes ; or — since tho means are equal — twice one of the moans is equal to the sum of tho extremes ; consequently, half tho sum of the means — or one of them, will be equal to half the sum of the extremes. Thus the arithmetical mean of 19 and (=23) ; and the proportion completed is 27 is 2 19 : 23 :: 23 : 27, for 19 + 2^=234-23. 20. If v/ith any four quantities the sum of the means is equal to the sum of the extremes, these quantities aro in arithmetical proportion. Let tlio quantities bo 8 7 5. As the sum of the means Ls equal to tho sum of tho extremes 8 + 5 = 0+7. Subtracting 6 from each of the equal quantities, wo have B+fv— 6 = 6 + 7— 6 ; and subtracting 5 from each of these, we have 8 + 5-6--5=6 + 7-6— 5. But 8 + 5=-6 — 5 is equal to .R — 6, since 5 to be added and 5 to bo subtracted are ?=;0 ; and +6 + 7—6—0 = 7—5, since 6 to be added an^l 6 to '/)Q subtracted =0 ; I 2 .82 PROPORTION, V' m tlierefore 8+5 — 6 5 R4-7 p, r. • .l fortion. It might in the same way be n-oyed Ui»i «y e^te ibnr quantities are in arithnfeacal^p^I^orUon 21. In A gamttrical proportion, "the jiroduct nf t^-'J fulfil ''' ^6* 8 IS a geometrical proportion, L~hv 7 2\ ' ^u tiplymg each of the equal quantl- nes by 7, we have (V»X7^ — 'Jv7. onri ^ u- i • .aehofthe.eby8,weUelfef6x'7(ox7''P'r^ •ut 14X8 13 the product of the extremes -and 1 6^7 B he product of the means. The same"eLonTn« lull r" rs^\ri;ter«-' '''»^-'-> -^^- Mt^a^-r^rpfet^^^^^^^^ 7X^S2-lVxnV iV f •^.- *''^g^°»«*™'J proportion, / X ^—HX n i and, dividing the equals by 7, we hayo 32 (one of the extremes) =1*^ (the product of the mo^ns divided by^the other extreme) ; and, dividing these by ll,wehaye-jj-(the product of the extremes di- vided by one mean)=14 (the other mean). We miBht l^l^Zli t 'l^' proportion would have answered just as well — and thfirfifm-o wKo* l__. _ -i . : in every case. ^ """" "'' "*'" ^^"^ ^ ^^^^ PROPORTION. ibH mo as 5, are itSS. Example. — Find a fourth proportional to 8, 10, and 14. Making the required quantity one of the extremes, we shall 10X14 Lave 8 : 10 : : 14 : ? ; and 8 : 10 : : 14 8 (the product of the means divided by the given extreme, ==17-5). And the proportion completed will be 8 : 10 : : 14 : 17-5. Making the required number one of the means, we shall 8x14 have 8 : 10 : : ? : 14 J and 8 : 10 10 (the product of the extremes divided by the given mean, =11-2) : 14. And the proportion completed vnll be 8 : 10 : : 11-2 : 14. £X£RCISf:8. Find fourth proportionals 1. To 8, 6. 6. 6, and 12 8 6, 12 10, 150 1020, 68 160, 10 68, 1020 68 150 1020 10 Jlns. 24. 16. 1020. 10. 68. 160. 24. If with any four quantities the product of the means is equal to the product of the extremes, these quantities are in geometrical proportion. Let the quantities be 5 20 6 24, As the product of the means is equal to the prod. !t of the extremes, 5x24=20x6. 5X24 20X6 Dividing the equals by 24, we have""^^ — = 24 ' J 5X24 20X6 and, dividing these by 20, we have 20X24 But::^ 5X24 _5 20X6 =20 5 and 20x24' 20X24- _6^ 5 _6 ^ =24' j therefore 20 ~ 24 * 20X24 consequently the geometrical relation between 5 and 20 two equal geometrical ratios — or a geometrical propor 184 PROPORTION. the pTodJotTelnZll «°™r'"f proportion, I eztromes. °^ '^ '^l""' '» ">« P'-oduot of the proportljfnal^""' '^'' '^™ '^ ™"y. '-> «■><• a fourth a{i;«._Knd the product of the second and third -Example, — Whnf i'<a +1,^ ^ ^ i-uira. 2^ ^ vvhat IS the fourth proportional to 1, 12, and W ^ • ^^ • •' 27 : 12x27=324 «nee dividing a nui fy "^u/dt^Tot'il?" u'^ '"^^- EXKRCISES. Find fourth proportionals ,X-Tol, 17, and 8 }?• » J' 23 „ 20 J|- » J. 53 „ 110 ^*»- •• I, 15 .. 1234 ^n*, j» 136. 460. 7300. 6830. 18510. by the first. ' ""^ "^ *''^°' ''iM' is not unity Ex.«P„.-Find a fourth prop„ra„„„ to 8, 1, „ud 5. 4-"^ut'it':ftL^;:etilV, "--."yt"" given product of both, ™hen tl'e otC hZu^ "JT^dered ^ tho pi-t,on .y „„H, p,„auce. no Sn^V?;- Sr'"" EXERCISES. Find fourth proportionals. fi on „.. 1 H. To 15. „ 16. ,. 17. 18. 19. 20. 21. 5. 6, 7, 8, 6, 37, ^?00, 1000 200, 1 20, and 1 21 24 1 1 1 20 1 1 50 68 1 1000 ^?i 4 4. 3. 8. Si. 4. 6. 6. = ' ^--"-^^or tiio extremes; and the proan^" ROLE OF PROPORTION. 185 of the extremes is equal to the mean multiplied by itself. Hence, to discover the geoinclriccl mexin of two quan- tities, we have only to find some number which, multi- plied by itself, will be equal to their product — that is, to find, what we shall terra hereafter, the square root of their product. Jhus 6 is the geometrical mean of 3 and 12; for 6X6=3X12. And 3 : 6 : : 6 : 12. 28. It will be useful to make the pupil acquainted with the following properties of a geometrical proportion — We may consider the same quantity either as a mean, or an extreme. Thus, if 5 : 10 : : 15 : 30 be a geometrical proportion, so also will 10 : 5 : : 30 : 15 ; for we obtain the same equal products in both cases — in the former, 5X 30=10 X 15 ; and in the latter, 10 X 15=5X30— which are the same thing. This change in the proportion ia called inversion. 29. The product of the means will continue equal to the product of the extremes— or, in other words, the proportion will remain unchanged — If we alternate the terms ; that is, if we say, " the first is to the third, as the second is to the fourth" — If we " mnltiplijy or dimk the first and second, oi the first and third terms, by the same quantity"— If we " read the proportion badcwards''^ — If we say " the first term plus the second is to the second, as the third plus the fourth is to the fourth"— If we say " the first term plus the second is to the fii'st, as the third plus the fourth is to the thii-d"— &c. RULE OF SIMPLE PROPORTT.ON. 3D. This rule, as we have sr.id, enables us, when threa quantities are given, to find a fourth proportional. The only difficulty consists in stating the question ; when this is done, the required term is easily found. _ In tlie rule of simple proportion, two ratios arc given, the one perfect, and the ether imperfect. 31. IluLE— I. Put '^^hut given quantity which belongs to the imperfect ratio in .u third place. II. If it appears from the nature of iho- question that the required quantity must bo greater than the other, J86 RULE OF PROPORTION. ■!!m ! 4%o~};^aY^^Tir^-^^^^^ 7-^3 fa wall in one It will faoiKrt Jb aLi Tth« n 'm *^^'T' ""^'^ ' question briefly, as fo lows-uini JZF^}^- ^^^^ ^^^'^ *''« represent the required qlLti^yl^ "^ ''"*' '^ interrogation to 5 men. 10 yards. 21 men. 'Jyai'da. ^r^^ps!z a;'iat ^■"'^^^^^^^ '^*^-^^ --^. P-/Jratio7an^d\^S3^,r"^*^*^^^ ^^»«h form the than 5 men the VpnmW "'i*^ ^ §'^'^*«'' ""'"^^r of yards than the^";, nu Jber~h«^^^^^^^ ^^'^' ^^" ^« S'-^ater term of the peSt ra^n ^n Ji,'° **"'' ^^ ^' ^° P"<= the larger the first plac?!I '^ '"^ *^° «^'°°'^' «^d *he smallerlu A ^ 5 ; 21 ; : 10 : ? And, completing the proportion, 5 : 21 : ; 10 • ^1 X 10 .o ^u . ■ ~~5 =^A *he required number. last 5 .en . m: tSt'^:^'Z,^g ?„/■- -8". i. to 3 men. 2 days. 5 men. *? days. tiitrbV^u^fiTher^ii'"^"^"' ""«'-^« ■»-'' „„ "'t^.'rs?,'' 'h" nnmbor of men, the shorter «.„ ,;„» „ .:..._ 4««nt..j, „, .,re,ui Will last thorn; but th«i',»7,^ «— g'Ji^ RULE OF PROPORTION. 187 required quantity — henco, in t'his case, the greater term of the perfect ratio is to bo put iu the first, tmd the smaller in Uio second place— 5 :3::2:'? And, completing the proportion, -=1|, tlie required term. 5:3::2 5 34. Example 3. — If 25 tons of coal cost £21, what will be the price of 1 ton '? 25 : 1 : : 21 pounds £.jp=lG5. 9-^-f/. 25 25" It is necessary in this case to reduce the pounds to lower denominations, in order to divide them by 25 ; this causea the answer, also, to be of different denominations. 35. Rkason of I. — It is convenient to make the required quantity the fourth term of the proportion — tliat is, one of the extremes. It could, however, be found eqxially well, if conHi- dered as a mean [23]. Urahon of II. — It is also convenient to make quantities of the same kind the terms of the game ratio ; because, for in- Btancc, wo can compare men with men, and days with days — > but wo cannot compare 7ne7>. with days. Still thero is nothing inaccurate in comparing the number of one, witli the number of the other ; nor in comparing the number of men with the quan- tity of work they perform, or with the nximbtr of loaves they eat ; for these things are proportioned to each otlier. Hence wo shall obtain the same result whether we state example 2, thus 6 : 3 :: 2 : ? or thus 5 : 2 : : 3 : ? When diminishing the kind of quantity which is in the per- fect ratio increases that kind whicli is in the imperfect — or the reverse — the question is sometimes said to belong to tlie inverse rule of three ; and different methods are given for the solution of the two species of questions. But liatton, in his Aritli- mctic, (third edition, London, 1753,) suggests the above gene- ral mode of solution. It is not accurate to say " the inverse rule of three" or " inverse rule of proportion ;" since, although there is an inverse ratio, there is no inverse proportion'. Reasoiv of III. — We multiply the second and third terms, and divide their product by the first, for reasons already given [22]. The answer is of the same kind as the third term, since iicitiici' tiic iiiullipliuukiuu, iiur aixxi UiTinluii Oi inia ici lii ixixa changed its nature ; — 20*. the payment of 5 days divided by 6 188 RULE OF PROPORTION. Of J da, ,„„u,p,iea b, 9 givo» ^- x as the pa,„o„t of u would not to the 4u^;a'';„;X;i^« «;»;,-<' tl,are.ore ij scc'oL,^«"fiLfa„°d'','i,i?,T"™\'" 'i'""^" *'"> «'■»' ""•' mo„ m'ca ure?wbe\ tW o""'' ''^ ""='■• ^■"■•"'=«' <«"■'- <itlier [29J. ^ '"'"" "'o ""'"poato to each Ex.„P..._,f 30 cwt cost ^24, what «„ 27 owt. co»t ' Dividing the first and second by d we have And, dividing the first and'third" by 4, EXERCISES FOR THE PUPIL. ^ ^^"J a fourth proportionalto 4. 6 yard, : I yard : : 27,, Am. 4,. Cul b. 5 lb 1 ib : : 155. ^,„ g^ 7. 4 yard, : iSyards : : u. Am. 4s. 6d. J? ^t Ssf *»^^^^^^ii^^^o to at £25 p„r p!ooe/eo:"?'Z."'^?!;;S "°^' ^2^. '«w much will 50 Ans. 121 „,"„(?„ ' '""° "■""" "'«y »"ffi» for 32 r ■ cwt"; 'aT. JIH^' '-'■ °f "-^J" -.St at 50. PC, «.o latter shall I ro.;iror''l:"2o";ardV'"" "'""'' "' the paymont payment of U ns the third, therefore it lio first and iatosfc coni- ite to each IS. £270 £23 per I will 50 i last 40 for 32 : 50.y. poj h wide, iiucli of RULE OF PROPOUTION. 189 , ,13. At 10.?. per barrel, what will be tho price of 130 barrels of barley ? Ans. £Q5. 14. At 5s. per lb, what will be the price of 150 ft) of tea ? Ans. 7505. 15. A merchant agreed with a carrier to bring 12 cwt. of goods 70 miles for 13 crowns, but his waggon being heavily laden, he was obliged to unload 2 cwt. ; how far should he carry the remainder for the same money ? Ans. 84 miles. lo. What will 150 cwt. of butter cost at £3 per cwt, } Ans. £450. 17. If I lend a person ^£400 for 7 months, how much ought he to lend me for 12 > Ans. £233 6s. 8d. 18. How much will a person walk in 70 days at tho rate of 30 miles per day .? Aiis. 2100. 19. If I spend £4 in one week, how much will I spend in 52 ? Ans. iS20S. 20. There are provisions in a town sufficient to sup- port 4000 soldiers for 3 months, how many must bo sent away to make them last 8 months ? Ans. 2500. 21. What is the rent of 167 acres at £2 per acre ? Ans. £334. 22. If a person travellmg 13 hours per day would finish a journey in 8 days, in what time will he accomplish it at the rate of 15 hours per day > Ans. 6|f days. 23. What is the cost of 256 gallons of brandy at 12s. per gallon ? Ans. 3072s. 24. What will 156 yards of cloth come to, at £2 per yard .? Ans. £312. 25. If one pound of sugar cost 8^Z., what will 112 pounds come to .? Ans. 896d. 2b. If 136 masons can build a fort in 28 days, how many men would be required to finish it in 8 days } Ans. 476. 27. If one yard of calico cost 6^., what will 56 yards come to ? Ans. 33bd. 28. What will be the price of 256 yards of tape ak 2d. per yard ? Ans. 612d. 29. If £100 produces me £6 interest in 365 days, what would bring the same amount in 30 da^'S .'' Ans j&i:<5iD iJi. 4a. '♦i ,f i m 14*0 nuLE OK I'noromio.v. 30. What shall I receive for 157 pair of gloves, at I Of/, per pair? Ans. 157 Od. fa > «*" 31 What would 29 pair of shoes como to, at 9* ner pair? Ans.2Gls. v., »t ^j. per aJ^: /- "^ ^TT ^'°^, ^"' neighbour a cart horse which draws lo cwt. for 30 days, how long should he have a • horse m return which draws 20 cwt'? Ans. 22i lys rr\l%a- "''"' P'\*^ ["^ '"*«'''''^ ^<^ -^'^ V^r cent." would give £6 m one month ? Ans. ^£1200 34 lfllendi2400for 12 months, how lone; our^htJEl. 50 be lent to me, to return the kindness ? Am. 32 months hsflO^orS' 'V ^^^^"^^"/'^re found sufficient to last 10,000 soldiers for 6 months, but it is resolved to add as many men as would cause them to be consu.hcd Anl 20 000. ' ''""'^'' '^ ™'" '""''^ '^'^ ^^"^ ^^ ^ /•n.^o ^^ ^,,^°7<^''' subsist on a certain quantity of hay for 2 months, how long will it last 12 horses ? A^^ 1} months. n/i^'/ '^^^P^^f'eper is so dislionest as to use a woi<rht of U for one of 16 pz. ; bow many pounds of just v be equal to 120 of unjust weight ? Ans. 105 lb rlnw • r' ""^ ^^^ **" ^^ '"^^^^^ ^J 40 men in 10 Ans.Vs^ days.""''^^ ""'^^^ '' ^' ^'''"^''^ ^^ ^^ "^^° •' are^Lt^IfTl *''' ^''^^'^ ''-'^"^^ ^'™' ^^^^^^ proportion are not of the same denomination ; or one, or both of them contain different denominations— HuLE.— Eeduce both to the lowest denomination con- tamed m either, and then divide the product of the second and third by the first term. pomrcoTt /-"^^ '^''" '''''''' '^ '''' '''' 1^^^- ^vh'-^t ^ill 87 The lowest denomination contained in either is ounces. ':■. ?_ '■ 1302X15 d. - ^ • j^J • • 15 : 3— =6960=£29. 1392 ounces. There is evidently the same ratio between 3 oz and 87 Th as between 3 oz. and 1392 oz. (the equal of 87 ft) RULE OF rnOPORTION. 101 ExAMPi.E 2.— If 3 yards of any thing cost 4^. 0J(/., what can 1)0 bought for £z i- The lowest denomination in either is farthings. s. d. 4 9? 12 57 ponce. 4 20 231 40 shillings. nls. 3. 231 farthings. 480 pence. 1920 farthings. There is evidently the same ratio between 4*. ^Id. and /2, fls between the numbers of farthings they contain, respectively For there is tlio same ratio between any two quantities, us between two others which are equal to them. Fa'amplk 3.— If 4 cwt., 3 qrs., 17 lb, cost XIO, how much will 7 cwt. 2 qrs. cost ? The lowest denomination in either is pounds. f' 840x10 19 : • ^^,^ =£29 Is. bd. cwt. qr. lb cwt qr. 4 3 17 : 7 2 4 4 19 ( ^rs. 30 < :|r8. 28 28 549 lbs. 840 K)3. EXERCISES. Find fourth proportionals to 39. 1 cwt. : 17 tons : : £5. Ans. £1700. 40. bs. : £20 : : 1 yard. Ans. 80 yards. 41. 80 yards : 1 qr. : : 4005. Am. Is. 3d. 42. 3s. 4d. : £1 10s. : : 1 yard. Atis. 9 yards. 43. 3 cwt. 2 qrs. : 8 cwt. 1 qr. : : £2. Ans. £4. 44. 10 acres, 3 roods, 20 perches : 21 acres 3 roods : £60. Ans. £120. 45. 10 tons, 5 cwt., 3 qrs., 14 ft : 20 tons, 11 cwt , 3 qrs. : : £840. Ans. £1680. 109 i.ja* RtTLE OK PROPORTION. cwf ? ^t' 'lltoo'"^° '' '' "^^ ^' '^"^^' '' "^^ P- pricL?i5irr'if ^-*« ''-> -Hat will bo tho yar^dt inri ^nn^^/ •Z"*^ '°'*' ^^ ^'- ^^^* ^iU no 17 .tf o °^*- ^^^^""er costs ^26 6^., how much 4ill ^7^ 8^1 cwt ^'' '"^"^ ''^''' can I have for £615 isi."? 57. How much beef can be bought for £760 12* al 1 ,1 i ^^' ^ ^''•' "^ ^^*-' cost £150, what will 3 ft, 1 oa., 1 1 dwt., cost ? Ans. £37 105. ' 69 If 10 yards cost 17.., what will 3 yards 2 ars cost? Am. bs. Uid. -» j'»iUH, -* qrs. 60. If 12 cwt. 22 ib cost £19, what will 2 cwt ^ qrs. cost ? Am. £4 5*. 8^^. "^ ^m ^ cwt. 3 n^nV 14 ^^ ""'•'/? ^7*-' ^^ g"'> «««* 19*-> what will 13 oz. 14 grs. cost ? Am. 15s. lOd. mination- ' ^^''^ *°''"' '°'''''*' °^ "'^''^ *^'"-^' ^'^« ^cno- if "^rf •■""■^fi!^"''^ '^*? ,*^^ ^^^^'^* denomination which contains then multiply it by the second, and div de the produc by the first term.-The answe^ wfll be of hat denommation to which the third has been reduced u^ rnay sometimes be changed to a higher [Se^ RULE OF I'ROPORTION. 19S Example 1.— If 3 yards cost ds. 21(1, what will 327 yards Tho lowest denomination in the third terri is farthings. yl"- ^i"- *• i\ 3'^7x441 £ s. d. 3 : 327 ; : 9 2| : ^ farthing8=50 1 6|. 12 ^)i. 110 pence. 4 441 farthings. Kx AMPLE 2.— If 2 yards 3 qrs. cost 11 W., what will 27 yards, 2 qrs., 2 nails, cost 1 Tho lowest denomination in the first and second is nails, and in the third farthings. yds. qr. yds. qr. n. 2 3 : 27 2 2 4 4 d. lU 442x45 — 4^ — farthing8=9<. bd. 11 qr. 110 qr. 4 4 44 nails. 442 nails. 45 farthings. Reducing the third term generally enables us to perform the required raultiplicatiou and division witli more facility. —It ia sometimes, however, unnecessary. Example.— If 3 lb cost £3 lis. 4\d., what will 96 lb cost? n> lb £. s. d. ^ s. d. £ s. d. £ s d 3: 06:: 3 11 4; : ' '^"^ =3 11 4Jx32=114 4 8 EXERCISES. Find fourth proportionals to 62. 2 tons : 14 tons : : ^228 10*. Ans. 199 10*. 63. 1 cwt. : 120 cwt. : : 18^. 64. Am. .£111. 64. 5 barrels : 100 barrels : : 6s. Id. Ans. £6 Us. Sd . 65. 112 ft) : 1 ft) ; : ies 10s. Ans. l{d. r66. 4 ft) : 112 ft) : : b\d. Ans. \2s. 3d. 67. 7 cwt., .3 qrs., 11 lb : 172 cwt., 2 qrs., 18 ft) : : £,3 9s. A\d. Ans. £87 55. Ad. n ;""!i 194 RULE OF PROPORTION. 68 172 cwt., 2 qrs., 18 lb : 7 cwt., 3 qrs., 11 lb : : ^87 6*. 3^^. A71S. £3 195. 4id. Am'Jl ^^^•' ^ ^^■^•' I'* * * 2 cwt., 3 qrs., 21 lb : : £73 70. £87 Gs. 3d. : £3 19s. 4-i^. . : 172 cwt., 2 qrs., 18 ib. Ans. 7 cwt., 3 qrs., 11 lb. > ^ ' 71 £3 195. 4irZ. r £87 65. 3r/. : : 7 cwt., 3 qrs., 11 lb. Am. 172 cwt., 2 qrs., 18 ib. > ^ » /l^^^'^tll^^^" ^'^' ^^"^ ^^*->^^a* ^ill 120 cwt. cost.? ^.L' £1^05*^4/" ''""'' ^^^''^"^ ''^ ^ ''^' ''' 74. What will 120 acres of land come to, at 145 6d per acre.? ^w5. £87. ' ' 75._ How much would 324 pieces come to, at 2s S^-d per piece ? Ans. £43 175. 6^/. ' f ' 76. Whafr is the price of 332 yards of cloth, at I65. 4^/. per yard .? ^7*5 £107 1 65. i/^" ■'•^ l?^^^^ ^^ ^^''"^^ ^^^^^ ^'' 4d!-) what will 18 lb 10 oz cost.? Ans. £49 135. 4d t. ?'L'. £1^2 13?4f ' "'^^ "'^ ' ^"^- ^ ^^- -- rent ( I 156 acres 3 roods .? Ans. £089 I45. ^fl" ^^* }^^' ^'^' P^^ ^''•' what will 56 cwt. 2 qrs bo worth.? Ans. £118 13.S-. ^ 81. At 155. 6^ per yard, wliat wHl 76 yards 3 qrs come CO .? Ans. £59 95. 7id ^ lb ?' 2' £r065.'' """'' ' "^ ''"' *'' ^* ^'' ''• P«^ 83 At 145. 4d. per cwt., what will be the cost of 12 cwt. J qrs. .? Ans. £8 195. 2d. 84. How much will 17 cwt. 2 qrs. come to, at 195. lOJ. -^er cwt. A71S. £17 75. Id. 2 n?; "^i"""*- °^^^"«^«osts £6 65., what will 17 cwt , 2 qrs , 7 lb, come to .? Ans. £102 125 lOi^^ ' ■;■ .^IJ ^^- ^'^ ^ cost ^^; lo.v. 9^ , Avhat will be the cost of DO cwt., 3 qrs, 24 Jb .? Ans. £378 I65. 8^1 RULE OF PROPORTION. 195 87. If tlic shilling loaf weigh 3 ft 6 oz., when flour sells at £1 13s. 6d. per cwt., what should be its weight when flour sells at £1 7s. 6d ? Ans. 4 lb 14f oz. i,£8. If 100 lb of anything cost .£25 Bs. 3d.,\lmt will be the price of 625 lb ? Ans. £WS 4s. 0-^~d. S9. If 1 lb of spice cost 105. Sc^., what is half an oz. worth > A'ns. Ad. 90. Bought 3 hhds. of brandy containing, respectively, Gl gals., 62 gals., and 62 gals. 2 qts., at Qs. Sd. per gallon ; what is their cost.? Ans. £Q1 16s. 8d. 39. If fractious, or mixed numbers are found in ono or more of the terms— • KuLE. — Having reduced them to improper fractions, if they are complex fractions, compound fractions, or mixed numbers— multiply the second and third terms together, and divide the product hy the first — according to the rules already given [Sec. IV. 36, &c., and 46. &C.J for the management of fractions. Example.— If 12 men build -3^ yards of wall in ? of a week, how long will they require to build 47 yards 1 Sf yarJs=2,6 yards, therefore . . i^X47_, 26 1 47 7 =9]- weeks, nearly. "0. — If all the terms are fractions — lluLE.— Invert the first, and then multiply all the terms together. ExAMPLK.~If f of a regiment consume \l of 40 tons of flour in | of a year, how long will ^- of the same regiment tako to consume it ? i'l--V- f Xl-T-|=^XfX?=,^=202-8 days. Tin's rule follows from that which was given for the division of one fractiou by another [See. IV. 49]. 41. If the first and second, or the first and third terms, are fractions-r— Kui.E. — llediuie them to a common denominator (should they not have ono already), and then omit tho denominatorsi i ' S^K i li(f \m 'Ji:li RULE or PROPORTION. a o^^t'cS"" ' "^ ^ ™'- "^ ""' «™'' •^2, what ,iU ^ of I : .J . . 2 : "J Reducing the fractions to a common "denominator, we have , , . fff •• U :: 2:? And omitting the denominator, 20:27::2:2^=£2-7=£2 14,. andVhiiH?®'"^^^^.""".^.*'^^^^"^ *^® fi'-^t and second, or the first BO^it^tTopitr"'^^" denominator-whiclfc^S^I Zll EXERCISES. cos'ti' ^Aii'ijr '"*' '"' "^^' ^"^ ' ^ ^^ -• aI%^''' """'^ ""^^ ^ ^"'^ '^^^ *« if 1 «ost is. ? of 'S^'^t^^'T^ ^^^^ ^^^-^ ^'^' ^« *^^ !>"- ^r'^7?f 4T]' '« ^^- «^ «^^-^ -«^ ^' 6i- per oz. .. I h'avo V^l^f^'^Tk^f-^ '^" "'^°^ P™^^ -« cosf ^7^«f i!^ *^ ?"'^ ^^ ^^T=V yards of cloth, if 7f cost i^7 lb5. 4d. .? ^W5. iE51 35. 113 3^ ' • R. ^??;7 ^l ^^^^.vi '^^^^ '' ^^'*^ ^981, what will ie363 85. 7^d. be worth .? ^?w. ^^^358 7, ij' ''^'* bought for ^i2'3''pP'l^ for 4| yards, how much can bo Dougttt tor £2j\ > Ans. 24 yards, nearly. MISCELLANEOUS EXERCISES IN SIMPLE PROPORTION. 102 Sold 4 hhds. of tobacco at 10ifZ ner TK • INTn 1 weighed 5 cwt., 2 qrs. ; No, 2- 5 ^™' V •_ P , ^« «- ^ iTirf 1 ^.. 1 /< K •»T lb ; and No. 4, 5 cwt., 1 ^j'., X i lu : ISO. pnoe.? ^7M £]04 Us. 9d , 1 qr., 21 lb. What RULE or PROPORTION. 197 1 03 . Suppose that a bale of merchandise weighs 300 Jb, and costs £15 45. 9d. ; that the duty is 2d. per pound ; that the freight is 255. ; and that the porterage home is Is. 6d. : how much does 1 ib stand me in ? £ s. d. 15 4 9 cost. 2 10 1 5 1 ft) 300 lb 1 duty. freight, 6 porterage. : 19 20 •iOT 12 1 3 entire cost. 300)4575 15|d. Answer. 104. Heceived 4 pipes of oil containing 480 gallons which cost 55. 5^d. per gallon ; paid for freight 45. pet pipe ; for duty, 6d. per gallon ; for porterage, l5. per pipe. What did the whole cost ; and what does it stand me in per gallon > Ans. It cost £144, or 65. per gallon 105. Bought three sorts of brandy, and an equal quantity of each sort : one sort at 55. ; another at 65. ; and the third at 75. What is the cost of the whole — one gallon with another ^ Ans. 6s. 106. Bought three kinds of vinegar, and an equal quantity of each kind : one at ^^d. ; another at 4d. ; and another at 4Ji. per quart. Having "mixed them I wish to know what the mixture cost me per quart } Ans. Ad. 107. Bought 4 kinds of salt, 100 barrels of each ; and the prices were 145., I65., 175., and 195. per barrel. If I mix them together, what wOl the mixture have cost me per barrel } Ans. I6s. 6d. 108. How many reams of paper at 95. 9<i., and 125. 3d. per ream shall I have, if I buy £55 worth of both, but an equal quantity of each .? An,s. 50 reams /\T an on 109. A vintner paid £171 for three kinds of wine : one kind wa,s £8 IO5. ; another £9 55. ; and the third 'if; lUS %!' RULK OF riioroRTroN. Iia.l of •fiJO l')s, nor hjii? TL. I,., j r» i ♦''-'--•'' orchid. ^^^^^^^^^^^^ 10 15 28 10 2H 10, the prioo (.f throo J.og.shoadH of oa.,h £ 171 , X171x;{ ■ £2H 10~^^ ^'''Js. I)arr,-I.s had I „f oach > yi,,, yoo ■*-'•"• ^'"'■f '"•■"ly weeks. "^^ weoiv ^ Ans. 56 provisions. Ifow lot' w^l/'^'^?^^^*'' ^^''^««« ^^^ ^>f and 2 ^^y^, ^ ^''"S ^'" *''^^3' ^^^ ^ ^ns. 26 weeks page. At .'],at |,,e n^I^ Te eVpo'ct'uo ho" '?''^^ copy contaming 400 pa/s P ^;.'^; %f;«"j;n « il^^^.y^^Z^7 'V^^^ "«'«b^r of 'each: 117. Suppose that a i^reyhoiiiul molroc 07 >vlnle a Jmre makes O;! ?nd h 1. • '^ 'P''"'S^ rqunl Icii.r(h T„ i„„; ,' ^"'^'^ '''^"' «P'"ings are of «^^'t.ikc.i, u .he IS au .prmgs before (he hound ? ' RULE or riiorouTiON. The tinio tukc^n by tlio gi-c^huuiid for ono that i-oquinMl by ihc hnw., ah 2') : 27 as 1 : n iJ9 Bprin/i; in to ^^^ U-J- 'I'J'u ^';r('yliomi(l, ilicn'iuio, iraiiiif "'.^p of OV U8 Hpnng Uunug ovcry ^prin;,' of (1h! luirc. 'Jlicrol oro tl»c : 50 : : 1 liavo will m snriii;^; : 5()-^^",=rzG7^), Ihc number of Hprinjra ako, bi'l'oro it is overtake:!. 10 118.^ If a tun of tallow oo.sts ,£35, nnd iH sold at tl . rate of 10 per cent, profit, what in tho solliinr prico > Ans. JU3S U)s. ° 119. If a ton of t;ilIo\v costs ,£.17 10.y., at what rntt muMt it bo Hold to gala by U) tons tlio price of I ton > Afis. £40. 120. JJought 45 barrels of boof at 21. v. per barrel; auion,!^ tliem aro IG barrels, 4 of whi(3h would bo wortii only li of tho rost. Mow muoli must I pay t Ans. £43 l.v. ^ -^ 121. If 840 oggs aro bought at the rate of TO for a penny, and 21G more at 8 for a peiniy, do I lose or gain if I soil all at ly for 2d. ? Avu. I gain ikl. 122. Suppose that 4 men do as much work as 5 women, and that 27 men reap a (juantity of corn in Mi days. In how many days would 21 wonum do it .? Ans. Tlio work of 4 mcn=that of 5 women. Thorefore (divldinj; each of tho equal quantities by 4, they will remain e(pialj^ 4 men's work . , s <hu work t)l'5 women -^ (one mans work )= - ■ ^ . Con- bcquently Ij times tho work of one woman=rl man's work .--^ that is, tho work of oiio man, in t(U'ma of a woman's woi-k, is 1{ ; or a woman's work is to a man's work :: I : 1'. Hence 27 mens work = 27xl| womon"s work 3 then, in place of Haying — 21 women : 27 men : : 13 days : ? say tho work of 21 women : the work of 27xU r=33n 3;J''xl3 ^- '^ %-=:20^« days. women : : 13.: _ 123. The ratio of the diameter of a circle to its circumference being that of 1 : .-J-Mlf)!), what is the circumference of a circle who.si! di;inu^ter is 47-3G feet ^ Ans. 148-78018 feet. 124. If a pound (Troy wrl-lit) of .silver i,s worih (JGs., !H\ 200 RULE OF PROPORTION. whaj^is the value of a pound avoirdupoise ^ Ans. ^ ere^ltttaX"^^ ^'^^^^ to his 16.. 35^. '''''^ ^'^ ^^^*- <^a" ie pay .? Ans. £m iei347 // t^^'feSl Z i"^"^"^^i^ «^ --1 costs Am. £1714 ihlUld ' *^' '°'* ^f ^« Irish mile.? 127. If the rent of 46 aprpq q ,.««j i >s JeiOO, what will be the re^t'of '^^' ^"^ " P^oh^s, 10 porches? Ans. ^vZTef, '' '"''"^' ^ ''^' ^ 12 mL aday^B wl!: Sf'" .f/"'^^ "' '""e rate of him. How iy mUea a dir^'i^'p*^ '^y'' "^''^"k both to have started C„ ,hfl ? "■*^<''- *"»™g 129. If the TOlue of Tn^ T° ^■^'f" ' ■^'"- 17. ^£4 0,. 2j/ how manvTir avou-dupoise weight bo Pomd1roy'>AmclS' '^y ^o tad & one anf S;e\wf:batt:»t\" ' ^^'"'"« '" "« ---n'! 'o' whaV : Se'w:!':?!:' \^'''™ "-- »» ^ jeiO 7s. 8J^. "'^ ""« ^^''ols garden? ^^j. ;> ef da^s raXrii-'dt" i:t ? '* ■^^^f ' « three do it ? A71S. 2~i2 ^ ^^^* *^^® ^o«ld all « ^a,s , 1 aa, , , ,-,,■ --^^^ ^^^^ _ ^ ^^^^ ^^^^^ ^^^^^ «i<^.- = lCa, = : lwi,o,o„ft.e;7k'r^fp!rntro&:; #4-3 1. 3 __i44 7 °^7^^<^ C would do in a day finished in ^ davk'Tn"!*!'';*. .">"»""=■■' '-'-^l •>« "We to do U by hbuself?" ^i^i. Jr^Y^'^ ""»"f « 1« ' ^ Atis. de4 1871 to his '^212577517 ? ^%*. ^£30 ' canal costs Irish mUe ? 14 perches, J roods, and the rate of 's, overtook 5l:i allowing 47W. 17. weight be id for one lis tenant ; Dies to £4 1 ^ Am. days; B would all le whole — 3 in a day. le whole — > in a day. e whole — > m a day. in a day. ivrork :: 1 le work) : Fit. B in 6^ '• will be d C be RULE or PROPORTION. 201 • ■ A, B, and C's work in one day=£ of the whole=|j|J Subtract- j A's work in 1 day=JV I _i i o of tha whole- **» ing j B's work in 1 day=/j. j -^s^ ^^ *^® wnoie-y^^, C'8 work in one day remains equal to . . . -^^^^ Then -f^-}^ (C's work in one day) : 1 whole of the work : : 1 day : 2 ^i|, the time required. 134. A ton of (Jbals yield about 9000 cubic feet of c;as ; a street lamp consumes about 5, and an argand Murner (one in which the air passes through the centre of the flame) 4 cubic feet in an hour. How many tona of coal would be required to keep 17493 street lamps, and 192724 argand burners in shops, &c., lighted for 1000 hours? Ans. 95373^. 135. The gas consumed in London requires about 50,000 tons of coal per annum. For how long a time would the gas this quantity may be supposed to pro- duce (at the rate of 9000 cubic feet per ton), keep one argand light (consuming 4 cubic feet per hour) con- stantly burning } Ans. 12842 years and 170 days. ? -* 136. It requires about 14,000 millions of silk worms to produce the silk consumed in the United Kingdom annually. Supposing that every pound requires 3500 worms, and that one-fifth is wasted in throwing, how many pounds of manufactured sill, may these worms be supposed to produce ^ Ans. 1488 tons, 1 cwt., 3 qrs., 17 1b. 137. If one fibre of silk will sustain 50 grains, how many would be required to support 97 tb } Ans 13580. • 138. One fibre of silk a mile long weighs but 12 grains ; how many miles would 4 millions of pounds, annually consumed in England, reach } Ans. 23333333331 miles. 139. A leaden shot of A\ inches in diameter weighs 17 lb ; but the size of a shot 4 inches in diameter, is to that of one A\ inches in diameter, as 64000 : 91125 : what is the weight of a leaden ball 4 inches in diameter > Ans. 11-9396. 140. The sloth does not advance more than 100 How loRij would it f o irn to IW! 1 f r im Dublin to Cork, allowing the distance to be 160 English mil©8 ? Ans. 2816 days; or 8 years, nearly. if 'i i „ I il ^:i\ 202 COMPOUND PROPORT/ON. 141. li'ugliish race horsea l.^vn i. i tl^oruto of 58 miles ai W t "^ '^ ^^ ^' "^ vclocify, „ii..i,^ *i,/ 1" 'loiii. In what time at n;. -i-ir'Xrt^ ^"^ll*'"'^ ^'* 3000 tons; '^^or^^lZti tii r-""" ^'rs'- ^''"ut the tlioii luto hair-smin J """i'^ny, mado into steel a,„l w-te, there aTo X3''Z"« /^'! '''''- "'"J-th" gnuus of steel? ^,„ Sjooa™ ''°° "'""" ™"3 COMPOUND PEOPOKTfON. -^^rfp^^^^^^^^^^^^ «. although t.o" i»'r."ft.t ratio „; th; thi.tt.erofl''''''"SH? '« «.e . jr. I'ut down the term, nf ? 1 . P™P""'''»n- ■"«.e first and second ace, if "V'" """^ ^••"™ antecedents may form one n!f ™'','' " ^V that the mother In ^ttin^ Zu 7,7' "I- "'" »™«eque„t: oflect It has upon thf ansZ-if „?""' ""^M" "hat '^' - ^-w man, JJ^ ^:^^^^'^^^;;J« ^ a wall in 20 icily i'^^^own{.32J,wiJlbea;a Hows m COMPOUND PRoroirnoN. 'J03 ^n to go at ^0, at til id Cork bo ^000 tons ; ' of 5 loet of 2 feet about tliG about ten pound of steel, and Joductiiic/ t>ut 7000 Jgli two stion, to ?>?• In ios, one to the I. ratios at the quents what idown anto- rm as livide in 20 I '3 • ir • k \ '^o'^^'tiwis which givo 2U days. 20 days imperfect ratio. 1 days, the number sought. 17 men 37 yards conditions which give the required number ol" Unys. 17 : 5 : : 20 : ? 10 : 37 And 17 10 5..oo.20x5x?>7 3y 17x10 'i'ho imperfect ratio consists of days — thoroforo we ar«i to (lilt 20, the given number of days, in the tliird place. Two ratios remain to be sot down — that of numbers of 7ncn, and that of numbers of yards. Taking the former first, wo ask ourselves how it affects the answer, and find tliat the more men there are, the smaller the required numlierwill be— tsinee the greater the number of men, the shorter the time ro{]uirod to do the work. We, therefore, set down 17 as anlecedeut, and 5 as consequent. Next, considering the ratio consisting of yards, we find that the larger the number of yards, the longer the time," before they are built — tlieroi'ore increasing their number increases the quantity retiuired. Hence we put 37 as consequent, and 10 as antecedent: and the whole will be as follows : — =13-0 days, n(;ariy. 45. The result obtained by the rule is the siinio .'is wonM lie found by taking, in succession, the two j)roportioa8 supposed by the question. Thus . 1 5 men would build 16 yards in 20 days, iu how )uany '8 woeld they build 37 yards ' .(3' : 87 : : 20 : ^" — number of days which 5 men would 16 require, to build 37 yards. 00 v37 If 5 men would build 87 yards in.tl_r2 — days, iu how many 16 days would 17 men build them ? 17 : 5 : : ?^ : 2^x5-17=20x5x37 ^^^ ^^^^^^ 16 16 17X10 of days found by the rule. 40. ExAMPLK 2. — Tf 3 men in 4 days of 12 working hours each build 37 perches, in liow many days of 6 working hours ought 22 men to build 970 perches '.* 204 22 8 87 8 :: 12 970 COMPOUND PROPORTlOJf. 3 men. 4 (iixyH. ■12 hours. 37 porches. ? days. 8 Jiours. 22 mon. 970 porches. " • " i'ui uiies. 3X12X97 0x4 22X8 X a/ """=21 i days, nearly. days^U^islhtlmport't ?.r """''^fore 4 place The more moZhXl^^ S.o^5^ '' P"' '" "»« ^'"rd form the work : therefore 29 jT * « *^''^*' necessary to per- smaller the nu^Cof worW E"* ^''?*' T*^ ^^ «««^""^I- K the number of days • hTnce 8^is Z fi *?' ^7' *^« ^'^^g'' The greater the number of perche?th^'*' ¥ ^^ «««""J- of days required to build tWn ^^'^ ^''^^^^ter the number put first, and 970 second ' consequently 17 is to bo or one in the first, a^ne in tL'"?! '^l? '^^"^ P^^«« J same number. ' '"^ *^^ ^^^''^ place, by tho Example 1 Tf ♦! 32 : IGO • • 8 • ^^^^X20x8 ^ 5 : 20 32x5 Dividing 32 and IfiO ^« qo , 1 1 5 4 8 : 5x4x8=100 measure a quantity in the S Z ''°^*?''' -"""^^^^ ^"' place ; or one in the first on?' ^"l^°o*her in the second This will in some iistanif 1 ^"°*^'' ^^ *^^^ ^^^'^ place into unity-^wSrrsrtr^^^^^^^^ ^-'^ti:- COMPOUM) PROPOKriON. 205 irly. therefore 4 n the third ^'^'•y to ncr- cond. The the larger 12 second, ho number 7 is to bo hy divid- id place ; J by tho iles Costa es cost ? uotients. pr{)|)or- !is long )er will second ' place a^ntities ExAMPLK 2— If 28 loads of Htono of IS^wt. each, build a wall 20 foot lon^ and 7 foot hip;h, how nian\ loads of lU cwt. wjll build one 323 feet long and 9 feet high ? : 28 : 15x323x9x28 _^^^_ 19 20 7 15 : 323 9 19X20X7 Dividing 7 and 28 by 7, we obtain 1 and 4.— Substitutine those, we have ° 19 : 15 : : 4 : 1 20 : 323 1:9 Dividing 20 and 15 by 5, tho quotients ore 4 aai 3 : 19 : 3 : : 4 : I 4 : 323 1 : 9 Dividing 4 and 4 by 4, the quotients are 1 and I : 19 : 3 : : 1 : ? 1 : 323 1 : 9 Dividing 19 and 323 by 19, tho quotients are 1 ,ind 17 : 1 : 3 :: 1 : 3x17x9=459. , . 1 : 17 1:9 In this process we moroly divide the first and second, or first and third terms, by the same number — which [29] does not alter the proportion. Or we divide the numerator and denominator of the fraction, found as the/oMr<A term, by the Kame number— which [Sec. IV. 15] does not alter the quo- tient. EXERCISES IN COMPOUNB PROPORTION. 1. If £240 in 16 months gains £64, how much will d£60 gain m 6 months ? Ans. £6. 2. With how many pounds sterling could I gain £5 per annum, if with £450 I gain £30 in 16 months ? Ans. £100. 3. A merchant agrees with a carrier to bring 15 cwt of goods 40 miles for 10 crowns. How much ought hi to pay, in proportion, to have 6 owt. carried 32 miles ) Ans. IGs. K 2 ii at U ii i^. :lf^«lf P ■^f)(j <o.ui.oi;nd phopoijtion. Am. £20. ^'^'^^J " ^'"'"'''^ lUO iiiilos » fo/q/^ f^^^ "' ''^ ^^^«»-cIiandl.so aro carriod 40 ,niln« for Js*^ itV/"^.^^'^""^^ "''^^''^ ^« -rried 60 S« fn. T 1^^^ * ^^ inorchandiso aro carried 9n ,«{i 1« honest™ Sr ie&AloO »r h' "'^* '" « ^'^^ would bo rcouimd f,^ 1 ''^ " ; """^ """V lioraM days ? AnXof ^'"'^ """^ *'"' """^'"'^ '« 3 bein« i J toThtCS^^^^^^^^^^ wag.' men's, and 24 paiv of won.onVIlmt' 1 P"'"' °^ each kind woiJd ic T 1 '''"'*'> V"" many pair of 13. A wall is r t,!' rl /■',"■ f women's shoes. how'^JI/iJrri/d'S i^ZT "''"T' u tunoei ,8 jays.^ .^„, X',^- tons the sa»o d.- < 10. 11 ^/j,-. are the wa.ws of 4 «,or, /•... - ^^ wages of i^sn iOf I COMPOUND PROPORTIOM. 307 what Jill bo tlio wages of 14 iiiea for 10 day^P Am. 16. If 120 busliols of corn List U horses 50 davs •7 • iu!^ ^^ '' ^?°^!''^" ^'"''^^^ ^^^ "»'^« i" 3 days when the days arc 14 hours Jong, in how many day«ot'7 hours each will ho travel 300 nriles ? Ans j^ " '^*' ^' ^ *'°"'« • i^'o^/ the price of 10 oz. of bread, when the corn ,s 4.. 2d per bushel, be 5^., what wm.st'be paid ibr 3 b 12 oz when the eorn is 5s. r,d. per bushel ? ^1... 3 . 3/. VJ. 5 compositors m 16 days of 14 hours lon^^ can compose 20 sheets of 24 pages in each she , 50^11 .t" days ot 7 hours long may 10 compositors compose a Bltr filV'"-'^^ ^"/^^^ """^ ^^^*«^' containi^ng 40 Bhects 16 pages in a sheet, 60 lines in a pa-e, and 50 letters m a line ? Avs. 32 days ° ' M^^p}^ ^'^' been calculated that a square degree (about 69X69 square miles) of water gives off by cvinor - tion 33 millions of tons of water |.er day. Ylow mu laLVt 7'\ ^^Vr^P^'^'"^ ''' ''^'ole surface to be 14 square feet ; and that the barometer stands at 31 inches ? Ans, 13 tons 19 cwt. QUESTIONS IN RATIOS AND PUOPORTIOX. 1. What is the rule of proportion; and is it ever called by any otlier name ? [IJ. 2 What is the difference between simple and com- pound proportion ? [30 and 421 3. What is a ratio ? [7]. 4. What are the antecedent and consequent ? [71 o. VV hat IS an inverse ratio ? [8] . 6 What is tlie difffironno betwn"r» "- -..:ii-_--.« 1 and a geometrical ratio ? [9j. ^^•^luat 1 r-1 208 COMPOUND PROPORTION. 7. ftow can we know whether or not an arithmohV^l or geometrical ratio, is altered in value p [10 and^' f'^ other? [Lj' "'"' '^''''''^'^ '^^'''''^ ^^ *^"^« «f ^"^ 9 What is a proportion, or analogy ? [Ul 10. What are means, and extremes ? flSl i.l%I^L:nir^^P'' - Soometial .oan of Jfc^roporL'rffe]''^' ^°« •!-»««- - - arith- miLKri^T™lf=" f- ■!-««. a.e in gco- fo^M'^llyZii^'" P"P°'"""^' '^ three quantUie, madt i!?T'"° the principal changes which may be made^m a geometrical proportion, without destroying r^li^;^"!? ^^ ?ecessarj, or even correct, to divide the rule of three into the direct, and inverse .P [35] 18 How IS the question solved, when the first o, second terms are not of the .same denomination • or one 19 hI-^''^ ''^*^^'^ ?^^^^^"* denomination ? [371^ if I' ll^Z '' ^ ^""'^^''^ ^" *^^« rule of proportion solved rf the thn^ term consists of more tha'n KeVoS .J^f' ^^"""^ i^i*' solved, if fractions or mixed numbe^-s oTin^rth^tet3%ra» ^^ ^"^-^^^ „f ^!' ^''° ^"^ °^ ""^ '''™« °f a question in the r„I„ 1 1 'ii f 209 A H I T JI M E T I C . PART II. SECTION yi. . PRACTICE. .t'»'SS jJis" '""'"•"s «.■ —.J Mie latter '' ^ '^ *^'' ^'""'^" ^« *« "le price of tlip ni-Loa '41 parts, and bndiaj' the sura of « nart, " ,;"/n- P"',"' •"■ ''y dividiug^tho price iZ ,>ot mJZTlZf ^^ a number, are those wf.ioh do any TateJ^ /o'^^t^'^idt: V-To'l T"''"^'^ "^ Mwe have seen fSeo It 2fiV,L f^?"' P"'''' ""' 3 To find ti,t „r T J' „ "■* "'"t^'' measure it. B „, . ^- • ? • T"' P"'^ of "ly number- ing Quofaflf.; \I ^l"?'' divisor, and the result. fmtjU ; :,;! rtd^^ct „re;!::r:\':e ;7r gte'n nurab:;:' "° ''^ ""'""""^ all «„t prrS'e " „' 810 PRACTICE. parts^f 'ST""""^^^'"^ "'' *^'° ^^'""^' ^^^ «o«^Pound aliquot 2)84 2>!2 ' 3)21 7)7 The prime aliquot parts aro 2, 3, and 7 ; and 2x2= 4^ 2x3= 6 2x7=14 2x2x3=12 [ "^''^ "'° compound aliquot parts. 2x2x7=28 2x3x7=4'^ 14:^21 *2ran'd42^'''^'' ^^""''^ '° '''^''' ^'' 2' ^' ^' ^' ^' ^2, ^ 5. Wo may apply this rule to appUcate numbers— Let it 2)240 2)jl20 2)G0 2)30 3)15 5)J5 2X2= • 2x3= 2x5= 2x2x2= 2x2x3= 12= 2x2x5= 20= 1 2x3x5= 30= 2 2x2x2x2= 16= 1 2x2x2x3= 24= 2 2x2x2x5= 40= 3 2x2x3x5= 00= 5 2x2x2x2x3= 48= 4 2x2x2x2x5= 80= fi 2x2x2x3x5=120=10 4 G 10 8 d. 8 6 4 4 8 in shillings PRACTICE. 211 And placed in order- £> d. 3 1 1!0 ¥(5 I 4^? ' = 4 ' = 5 tV= 16= 1 it d. 4 1 8 24;= 2 V= 8 2^=12=1 d. 1= 30= 2 6 1= 40= 3 4 i= 48= 4 i= GO 5 A= 80= G 8 I i=120=10 Aliquot parts of a shilling, obtained in the same way- 1 1 T? — T _l I 2? 2 _l J s. d. Aliquot parts of avoirdupoise weight— s. d* 4=6 Aliquot parts of a ton. ton cwt. or I ; 1 9 ?o — 2 — ■^ 20 ■■• ^ TIT— ^X= «5 tV= 2 = 8 J= 2.!=10 ^= 4"=16 ^= 5 =20 J=10 =40 Aliquot parts of a cwt. cwt. K) '=2 ! 2? ■» = 7 8 T? i=14 4=16 i=28 1=56 Aliquot parts of a quarter qr. ib A=2 1=14 , Aliquot parts may, in the same manner, be easily obtamed by the pupil from the other tables of weights and measures, page 3, &c. ^ 6. To find the price of a quantity of one dcnomina- tion— the price of a " higher" being given Rule.— Divide the price by thai number which ex- presses how many times we must take the lower to make the amount equal to one of the higher d'enomina- pei^^cwrr'~^^''* '' *^^ P''"' ^^ ^"^ ^^ ^^ ^"**^^ ^t '2.. Tl^^f'''"!*^^^^^ ^\^^ ^ «*""^ « ti"^es, to make 1 cwt. Therefore the price of 1 cwt. divided by 8, or 72s -^S-oI IS the price of 14 ib. ^ , ux <^. . o—y*., The table of tiliquot pn'^^ of avoirdimms,> ^xo^r-ht -hr^s, ti;f pile'' o'F l*ti"'' " ^' "• ^■'■'•-f"-"'" F- i*^ '!>; J rf^ lis ill 212 PRACTICE. 1. 2. 3. 4. EXERCISES. What is the price of i cwt. at 29s. 6d. per cwt. ? Ans. Is. 4^d ,- a yard of cloth, at Ss. Gd. per yard ? An!. 4s. 3d 14 ft, of sugar at 45.. 6d. p.r cwt. ? Ans. 5s. S^d What IS the price of ^ cwt., at 50.. per cwt. > £ s. d. 505.=2 10 qra. cwt. The price of 2=i is of 1=1-^.2 IS ] £> s. 0=2 10-^2 12 G=l 54-2 5 . Therefore the price of 2-f 1 qrs.(=| cwt.) is 1 i7g J« half the price of 2 qrl'' Th&.rth " pi^'e'of I'llTu onf ow" ^"'' '' ' '^'- P^"^ *^- ^^^ «f '^"f tlfe ^ile of 6. 6. 7. 8. . What is the price of ? oz- of cloves, at 9.. 4rf. per lb ? Am. 3id nail of lace, at 15.. 4d. per yard ? Ans."" ' 2 10, at 23s. 4d. per cwt. .? yl?^.?. ii^ ^ ib, at 18.. Sd. per cwt. ? Ans li^." 14^/. 7. When the pric3 of wwrc ^A^w o^j^ 'qow->r'' dono minatiou is required— ^^^ Ia«f vnf;""^'^^^^' P""' °^ '^^^^^ denomination by the last rule ; and the sum of the prices obtained will be the requu-ed quantity. [ '^^ at «;.;•:? c-t'"?^' " "" P™= °f 2 <!-■ 1* » of -gar, s. d. .45 price of 1 cwt. cwt A-nAoo A ti- ^ . [or ,1 of 1 cwt. 14\b=f,or'-of2or8 ^■^' ^^ ^•^••-^/=22.. G,i.4-4, is the s, or , 01 ^ qrs. p^ce of 14 lb, the i of 1 cwt.. \ ^^i^~T; . 01^ the { of 2 qrs. 2nr« t n . i H IS the price of 2 qrs. 14 lb. ^ ^J^«-rT5 ?f 1 cwt. Therefore 45.^ (f,li« r^vj^. of 1 o.-f ^ - o or zos. Qu., 18 the price of 2 qrs, ^' ' o^,t.J~2, PRACTICE. 213 Ans. 4s. 3d Ins. OS. S^^d r Gwt. ? d. £ s. 0=2 10-^2 _G=1 5h-2 G \vt. : md ita nd its price of f cwt. is lie i^rice of . S^d. ns. llid. 3r" dcno- on by tlie d will be of sugar, of 1 cwt. Bof2qrs., ^4, is the of 1 cwt., 14 lb. nTt.)-f2, 45^^ JoJ f, ^ rV ^i^.y^^i^f 2 qrs. Therefore T rJo ? ; f?'- S'f7-'^=5*- 7Arf., is the price of 14 !b. f; mT. • l"^^'--^^'^-'.^.' *^^ P'^^*'^ ^f 2 qrs. plus the price ot 14 lb, IS the price of 2 qrs. 14 lb. i i- f EXERCISES. What is the price of 9. 1 qr., 14 fe at 46^. 6d. per cwt. ? Ans. 17s. 5id. 10. 3 qrs. 2 nails, at 17*. 6c?. per yard ^ Ans los. 3|r/. 11.5 roods 14 perches at 3s. lOd. per acre ? Ans. 5s. l^d. 12. 16 dwt. 14 grs., at £4 4s. 9d. per oz. ? Ans £3 10s. 3}d. ^ 13. 14 lb 5 oz., at 25.5. 4d. per cwt. ? Ans. 3s. 2fd 8. When the price of ow, "higher" denomination is required — KuLE. — Find whaf ninnbor of times the lower deno- mination must be taken, to make a quantity equal to one of the given denomination ; and multiply the price by that number. (This is the reverse of the rule eiven above [G]). ^ ExAMPLK.— What is the price of 2 tons of sucar, at 50s. per cwt. < 1 \?.^'^'^' ^^ *^® ^'« ^^ 2 tone : hence tlie price of 2 tons will be 40 times th price of 1 cwt.— or 50,9.x40=£100. 50.S-. the price of 1 cwt. multiplied ^y 40 the number of hundreds in 2 tons, gives 2000,s-. or XlOO as the price of 40 cwt., or 2 tons. EXERCISES. What is the price of 14. 47 cwt., at l.y. S^. per lb I Ans. £438 VSs. 4d 15. 36 yards, at 4d. per nail r Ans. £[) V2s. 16. 14 acres, at 5s. per porch .? Ans. £5f)0. 17. 12 R), at l|f/. per grain ? Ans. £504. IS. 19 hhds., at 3d. per gallon : Ans. i214 19*. 3^. 0. When the price of more Ihitu one "higher" dcno- miuatiou is required — ill 214 PRACTICE • RuLE.—Find the price of each bv the lasf nn^ nrU atlTr"o"^eT^* " *'^ ^'^^ <^^ ^ «-*• ^ <1- of flour, 1 stone is the j\ of 2 cwt. Therefore tv,„u- r ^ 1. ,?^'-'*^o price of one stone, ' multiphed by B, the number of stones in 2 cwt, gives 3a?., ;• ^?ce ot 16 stones, or 2 cwt. EXERCISES. in e , What is the price of .£1 L ^ ' ^ ^''•' ^ °^"'' ^* ¥' P«^ ^^« -^ ^^^. 20. 6 cwt. 14 ib at 3^. per ib ? Ans. £8 Us. 6d. ^1. 3 ib 5 oz. at 2id. per oz. ? Ans. 9s. lUd. ul^fieMTlfc.' ^^^^«>3P-^^-> at 5. per perch .P fini^/i. ^^^"^ *^.^ P"^^ °^ ^"^ denomination is given to find the price of any number of another— ^ ' ±tuLE.-.Find the price of one of that other denomi- nation, and multiply it by the given number of the ^^XAMPLE.-What is the price of 13 stones at 255. per 1 stone=| cwt. Therefore 8)25^, t he price of 1 cwt. divided by 8, we obtain £2 7^ M the price of 13 stones. Istoneistliejoflowt. Hence 25.!.^8=3s 1 u i, th. pnoe of one stone; and 3,. lirf.xlS, the price of is '.it! PRACTICE. 816 EXERCISES. What is the price of 24. 19 lb, at 2d. per oz. .? Atis. £,2 \0s. Sd. 25. 13 oz., at \s. Ad. per ft) ? Am. \s. \d. 26. 14 ft), at 2s. Qd. per dwt. ? Ans. &420. 27. 15 acres, at 185. per perch ? Ans. ^£2160. 28. 8 yards, at Ad. per nail ? Ans. £2 2s. Sd. 29. 12 hhds., at 5d. per pint ? Ans. ^£126. 30. 3 quarts, at 91^. per hhd. ? Ans. Is. Id. 11. When the price of a given denomination is the aliquot part of a shilling, to find the price of any num- ber of that denomination — Rule. — Divide the amount of the given denomina- tion by the number expressing what aliquot part the given price is of a shilling, and the quotient will be the required price in shillings, &c. Example. — What is the price of 831 articles at 4d. per } 3)831 277s.=£13 17s., is the required price. Ad. is the | of a shilling. Hence the price at Ad. is i of what it would be at Is. per article. But the price at Is. per article would be 831s.:— therefore the price at Ad, is 831s. -^ 3 or 277s. lii EXERCISES. What is the price of 31. 379 ft) of sugar, at 6d. per lb ? Aiis. £9 9s. 6d. 32. 5014 yards of calico, at 3d. per yard.? Ans. £62 13s. 6d. 33. 258 yards of tape, at 2d. per yard ? Ans. £2 3s. 12. When the price of a given denomination is the aliquot part of a pound, to find the price of any number of that denomination — Rule. — Divide the quantity whose price Is sought by that number which expresses what aliquot part the given price is of a pound. The quotient will be the re- quired price in pounds, &c. 216 PRACTICE. ^ J«Mr,..._What is tI.o price of 1732 ft of to., .t 6, would bo jeiT-^p fi. r P°.^ ^^- ^^^^ at ^1 per lb it EXERCISES. Wliat is tho price of 35. 13 oz., at 4.. per oz. ? Ans. £2 12s. 37. 83 a, at Is. Ad. per ib ? Ans. £5 10s Sd 39* 976 r-V'f o'^- P°^ ^^- -^ ^-- ^3 16.. si. 4?' "? »»' ^* ^^>^- pel- ft ? ^.^..^2 6.. 8^ 44. 1000 ib, at 35. Ad. per lb .? ^w.. jgieg 13,. 4J. 13. The complement of the Drice i«i wliof ;f ,„„ * i. pound or a shilling ^ ^* ^* ^^^*^ ^^ » por^."d r-"^''"* '•'"'» Prf^o "f 1«0 yard., at 13», «. Cs. Sd. (the complement of 13s. 4^-) ia 1 of £1 Z" f 'S *■>» Pri"* »t *1 per yard .uhtraot ^, the ^rioo at fe. sS. (tCoomplement) andthedi£fe™oo, 980, will ho the price at ^^Spor yard. I47^™tTlf\t]^:.1;:;' t'J^O at C. 8. are e,„al to price of 1470 at^lSs, «34hepr1oe^ofl?70 at £?'"'•*''' .the price of 1470 at Gs. &(. per yard ' '°"" PRACTICE. 217 ..■i tea, at 5j 2 lb is tht, 1 per lb it > 135. 4d. 12 7s. 6d. Sd. 8d. 3 9*. 2d. • 13*. 4d. tnts of a uot part lot — ho caso IcuJated 13s. 4cl 1. lement) 3r yard. qual to ice the I minus EXERCtSEiS. What is tho price of 45. 51 ib, at 175. 6^. per lb .? Ans. £44 125. 6a 46. 39 oz., at 7d. per oz. ? Ans. £1 2s. 9d. 47. 91 ft), at lOd. per ib ? Ans. £3 15s. lOd. 48. 432 cwt., at 165. per cwt. } Ans. £345 12s. 14. When neither the price nor its complement is the aliquot part or parts of a pound or shilling — Rule 1. — Divide the price into pounds (if there are any), and aliquot parts of a pound or shilling; then find the price at each of these (by preceding rules) : — the sum of the prices will be what is required. Example.— What is tho price of 822 lb, at £5 19s. SJtZ. per lb I £5 m. ^d.=£6+m. Zld. s. d. £> 8 —I !fM But 195. ZM. < 10 G 2 0=1 or jV of the last or ^ of the last Henco tho price at £5 195. Z'^d. is equal to £ 822x5 82a £ s. d. =4110 0, tho price at 822 3 8 22 It )l22?^123_i_ A\ = 411 = 274 = 102 15 = 5 2 9 = 17 14 £ s. d. 5 per lb. £i or 10 £l or 6 8 £] or 2 6 ;; £ I ov 1-J ;;4i;oro ooi 11 11 And X4903 14 lOi is the price at £5 19 3 J „ The price at tho whole, is evidently equal to the sura of the prices at each of tho parts. If the price were £5 19s. 3^d. per lb, we should sub- tract, and not add the price at }d. per lb ; and wc then would have £4902 05. l^d. as the answer. 15. Rule 2.— Find tho price at a pound, a shilling, a penny and a farthing ; then multiply each by thoir [ , II f '! t iJI n m^:^' 218 PRACTICE. produl'' Tf""^' m' '" '^' Sivon piico ; and add the pioaucta. Using the same example £ ». d. £ a ^llui S S5j;»®P^!«eat£l)x 5=4110 6 Its Q X (*?0P^'ceatl«.)xi9= 780 18 ^ 1? J.Sf'oP'-Jceatle/OX 3= 10 5 i7 li(tho price at 4flf.)x 8= 2 11 d. the price at £5 » 19*. 6 » 8d 4i „ id. And the price at £5 Ids. 3|rf. is ^£4903 14 lOi tlie^hi.w''/'"-^^^. *^' P^'^°" ^' *»^« "«^t number of the highest denomination ; and deduct the price at tlia difference between the assumed and given prFce Usmg stiU the same example— prfco.'' '''''' *' ^^-'^^ ^'^^''^ denomination in the givea From the price at £6 ^ "' ^' nr 4?qo n n Deduct the pri^e( the price at 8rf.=27' 8 > ^ ^ ^ "'*'''' < .. 4rf.= IZlijO' 28 5 U at 8id. The diflFerence will bo the price at £6 19*. 8| or i^lTm 17 Rule 4.— Find the price at the next hijrher atT <^ff ' '^ ' Tf^^ '' ^""^"g ' ^-d deduct the p1:Se at the difference between the assumed, and given price ExAMPLE.-What is the price of 84 lb, at Qs. per tb^ ds.-^Gs. 8d. minus 8J.=^ minus i-^10. we have ^25 4 0, the price at 6s. EXERCISES. ^r. ^o « ^^^^'^^ ^^ *^^ P^ice of 49. 73 ib, at 13^. per ft ? Ans. £^7 9s. i>0. 97 cwt., at 15^. Qd. per cwt. ? Ans. £1Q 7s. 51. 43 ft, at 3s. 2d. per ft ? Ans. £6 'l6s. 2d 9d. r)2. 13 OS. lid. 53. 27 yards, at 7*. ' 5U. Is. llid. acres, at £4 5s. lid. per acre.? Am. £55 per yard.? Ans. JBIO llsTtl t' p™' '^ "" "™ °""»''^' °f ^'i»S». PRACTICE. 219 add tlio >rioe at £5 „ Ids. 8d id. imber of 30 at tlid lie given a. i2 d !8 5 li 3 14 lOi higher he price n price pep lb. 7.1. 9d. d. . £10 illings, Rule. — Multiply the number of articles by half the number of Hhillings ; and consider the tens of the pro- duct as pounds, and the units doubled^ as Hhillings. Example. — What i{ the price of G4G lb, at 16s. per lb 1 646 8 510 £510 10,s. 2s. being the tenth of a pound, there are, in the price, half as many tenths as shillings. Therefore half the number of shillings, multiplied by the number of articles, will express the number of tenths of a pound in the price of the entire. The tens of these tenths will be the number of pounds ; and the units (bcigg tenths of a pound) will be half the required number of shillings— or, multiplied by 2— the required num- ber of shillings. In the example, 10.9., or £-8, is the price of each article. Therefore, since there are 040 articles, 040xi^8=ii5.10-8 is the price of them. But 8 tenths of a pound (the unitHn the product obtained) , are twice as many shillings ; and henco we are to multiply the units in the product by 2. EXERCISES. What is the price of 54. 3215 ells, at 6s. per ell.? Ans. £964 lO.v. 55. 7563 lb, at 8s. per lb } Ans. £3025 4.-. 56. 269 cwt., at 16s. per cwt. } A"'!. £215' 4s. 57. 27 oz., at 4s. per oz. .? Ans. £ 8s. 58. 84 gallons, at 14s. per gallon ." Ans. £58 I6s. 19. When the price is an odd number of shillings, and less than 20 — Ruj^E. — Find the amount at the next lower even number of shillings ; and add the price at one shilling. Example.— What is the price of 275 lb, at 17s. per lb ? 8 220 13 15 The price at 16s. (by the last rule) is The price at Is. is 275s.= Hence the price at 16s.+ls , or 17s., is £233 15s. i Ik Ih 220 PnACTICB Tho prico at 17.v. is equal to tlio prioe at oiio aJjiliing. price at IGs., plus the EXERCISES. 59. 86 oz., at 5*. per oz. ? Ans. £'21 10.? 60. 62 cwt at 195. por cwt. ? Ans. £i)S ' ISs. ro ^^yf^^'^^ 175. per yard.? yl^i^. X'll is*. 02. 439 tons, at 11^. per ton ? Ans. jC;241 9.v. Od. 96 gallons, at 7s. per gallon ? Ans. i233 12^. number-'''' *^° ^""""^'^^ '' ^^P^-^^^^^^^'l l>y a mixed liuLE.— Find the prico of tho integral part. Then t7o"n t^d d- ^'ny^'' ^ «- nume?ato/of the fr J. ti^on and divide the product by its dcnoniinator-tho quotien will be the price of tho fractional part. To sum of these prices will be the price of the whole quan- ^ ExAMPLE.-What is the price of ^ lb of tea, at 5.. per The price of8 lb is 8x5?.= The prico of |- lb is ^^'• £ f. (I 2 3 9 And the price ofS^ lb is . . 2~T~U of a EXERCISES. 64. 65. 66. 67. 4id. 68. 4s. ed. 69. iP2751 What is the price of oIqIT""' ?* ^^- ^f' P*^^ ^°2«" -^ ^^^•'. 175. 101./. xi/dy It), at 25. brf. per lb .? ^^5. ^£34 3^ ] j^" 5302 lb, at 145. pei: lb ? Ans. 371 IO5. 'e/ ' 178f cwt., at 175. per cwt. ? Ans. i2151 125 7023 cwt., at £1 12s. 6d. per cwt. .? Ans. £1239 ^\^^\.,?*-' ^^ ^^ ^'- '^^' per cwt..? Ar^. lis. b^d. PRACTICK. 221 s.f plus the lS,s. 18*. 9.V. !3 12s. a njixcd t. Then tho frac- %tOT — tho rt. Tho Die quan- at 5.'?. i)cr -)v\co of a s. 101(1. '. Ud. 6(L 51 125 ^1239 ' Ans. 21. The rules for finding tho pnco of Bovoral deno- minations, that of one being given [7 and 9], may bo abbreviated by those which follow — Avoinliopoise Weight. — Given the price per cwt., to find the price of hundreds, quarters, &c. — lluLE. — Having brought tho tons, if any, to cwt., multiply 1 by tho number of hundreds, and consider tho product as pounds sterling ; 5 by the number of quar- ters, and consider tho product as shillings ; 2^, tho number of pounds, and consider the product as pence : — tho sum of all the products will be tho price at £1 per cwt. From this find tho price, at the given. number of poun '''=', shillings, &c. , Example.— What is the price of 472 cwt., 3 qrs., 10 lb, at £5 \)s. 6(1. per cwt. 1 £ s. d. I 5 2'- Multipliers 472 3 16^ 472 17 10| is the price at £1 per cwt. 5 2589 1 9| the price, at £5 9s At £1 per cwt., there will be £1 for every cwt. We mul- tiply the qrs. by 5, for shillings ; because, if one cwt. costs £1, the fourth of 1 cwt., or one quarter, will cost the lourth ot a pound, or 5s.— and there will be as many times 5s. as there are quarters. The pounds are multiplied by 2\ ; because it the quarter costs 5s., the 28th part of a quarter, or 1 lb, must cost the 28th part of 5s., or S^c/.— and there will be as many times ^d. as there are pounds. EXERCISES. What is the price of 70. 499 cwt., 3 qrs., 25 fib, at 25s. lid. per cwt. ? Ans. £647 I7s. l\d. 71. 106 cwt., 3 qrs., 14 ft), at 18*. M, per owt. ? Ans. jeiOO 35. 10|rf. PRACTICE, ^I'-^mo 55'.'4.r- ' '^' =" '''■ '"■' P«' -'• ? jB4?i7l° eZ*'' ^ '■'''"■' '* '''' *' '*'• "''• P""' •='"■ ■' ^'"• jes^ss^sij''' ^ '^^^'' '' "^' "' ''™- "''• P" ™'- • ^'"• A«:£L r iwf/"' '" *' ^' '^'- '"■ p='- -'• •' ^r-iri 7^/ '■■■' '' ''' "* '''■ ^- p°'- -'••' 22. To find the price of cwfc., qrs., &c., the price of a pound being given — RuLE.—Haviug reduced the tons, if any, to cwt., multiply 9. Ad. by the number of pence contained in the price of one pound :— this will be the price of one """^i?"!. I^^ *!'^ P"^"-' °^°"^ ''^^- ^'y 4, and the quotient Will be the price of one quarter, &c. Multiply the price of 1 cwt. by the number of cwt. • the price of a quarter by the number of quarters ; the price of a pound by the number of pounds ; and the sum o± the products will be the price of tlie given quantity. Example. 8(/. per lb. ? d. -What is the price of 4 cwt., 3 qrs., 7 lb, at 9 4 8 s. d. 28 R « f ^ •'' ''H '"^^- Xj.^i" give 298 8 the price Of 4 cwt. 28)18 8 e price of Iqr. X3,willgive 66 the price of 3 qrs. 8 the price of 1 lb X7, will gi ve 4 8 the price of 7 lb. 20)369 4 And the price of the whole will be £17 l£~i At Id. per lb the price of 1 cwt. would be l\M. or 9i-. Ad. •— therefore the price por cwt. will be as many times 0*. U as there are pence in the price of .a nnnnd. The >iri-" -p - quarter IS \ the price ot 1 cwt. ; an<l there will bo as many times the price of a quarter, as there are quarters, &c. PRACTICE. 223 !l»'ifl EXERCISES. What is tlie price of 79. 1 cwt., at 6d. per ft) > Am. £2 16s. 80. 3 cwt., 2 qrs., 5 ib, at 4d. per Bb .? Atis. £6 12s. 4d. 81. 61 cwt., 3 qrs., 21 lb, at 9d. per Bb } Am. £2\B 2s. M. 82. 42 cwt., qrs., 5 ft), at 2bd. per lb r iln*. ^2490 105. bd. 83. 10 cwt., 3 qrs., 27 Bb, at bid. per ft) ? Am. £2Q\ Us. Qd. 23. Given the price of a pound, to find that of a ton — EuLE. — ^Multiply £Q 6s. 8d: by the number of pence contained in the price of a pound. Example. — ^What is the price of a ton, at 7d, per ft) '? JS s. d. 9 6 8 7 65 6 8 is the price of 1 ton. If one pound cost Id., a ton will cost 2240rf., or £9 6s. Sd. Hence there will be as many times £9 6s. Sd. in the price of a ton, as there are pence in the price of a pound. EXERCISES. What is the price of 84. 1 ton, at 3^. per ft) > Am. £28. 85. 1 ton, at 9d. per ft) .? Am. £84. 86. 1 ton, at 10c?. per ft) ? Am. £93 6^. 8d. 87. 1 ton, at 4d. per ft) T Am. £37 65. Sd. The price of any number of tons will be found, if we mul- tiply the price of 1 ton by that ntmiber. 24. Troy Weight. — Given the price of an ounce — ^to find that of ounces, pennyweights, &c. — Rule. — Having reduced the pounds, if any, to ounces, set down the ounces as pounds sterling ; the dwt. as shillmgs ; and the grs. as halfpence : — this will give the price at £1 per ounce. Take the same part, or parts, &c., of this, as the price per ounce is of a pound. 224 PRACTICE. Example 1 -What is the price of 538 oz., 18 dwt, 14 grs., at lis. 6d. per oz. ? » , ^^ uwt , i^ Us. Gd. =il-L.^J4.ij^2 £> s. d. 2)538 18 7 is the price, at ^1 per ounce. ^ol^fr 1 « i?f ^' *^® P^^^«' a* 10^- per ounce. ^' ^ 1 Q n li }^ *^® P"^®' «-* !«• per ounce. •^^ '^ 5f 18 the price, at 6d. per ounce. And 309 17 8i is the price, at lb. 6d. per ounce. 14 halfpence are set down as 7 pence. If one ounce or 20 dwt. cost £1, 1 dwt. or the 20th part -^4th part of 1 dwt., or 1 gr. will cost the 24th part of IS. — or ^o. ^ ^E^xAMPLE 2.--What is the price of 8 oz. 20 grs., at .^£3 £> s. d. 8 10 is the price, at £1 per ounce. o Price It £1 ' 10-0 ifi ? • t5^ P"^®' ""^ 5^ P^^ ^^''e- vTnl I o A n . 1. i^ *^e P"«e, at 2s. per ounce. I nee at 2s.^ 4 =0 4 0^ is the price, at 6d. per ounce. And £25 2 7^ is the price, at ^£3 2s. 6(i. per oz. EXERCISES. What is the price of 88. 147 oz., 14 dwt., 14 grs., at 75. 6d. per oz. .? -4715. £55 7*. 11^^. p "^.. , 89. 194 oz., 13 dwt., 16 grs., at 11*. 6d. per oz. ? Ans. ieill 18*. 10^(£. ^ 90. 214 oz., 14 dwt., 16 grs., at 12*. 6d. per oz. } Ans. £134 45. 2d. ^ 91. 11 ft), 10 oz., 10 dwt., 20 grs., at 105. per oz. ? Ans. £71 55. 5eZ. "^ 92. 19 ft), 4oz.,3grs.,at£2 55. 2(^. peroz. .? ^W5. £523 185. lli<^. ■^ 93. 3 oz., 5 dwt., 12 grs., at £1 65. 8<^. per oz. ? Ans. £4 7*. 3J</. llHi ' ^i PRACTICK. 225 18 dwt, 14 ice. ice. ce. je. ' ounce. 3 20th part . ; and tlio th paxt of 5rs., at .-£3 r ounce. per ounce, per ounce* per ounce. Qd. per oz. per oz. ? per oz. ? per oz. } per oz. ? ? Ans. per oz. ? 25. Cloth Measure. — Griven the price per yard — to find the price of yards, quarters, &c. — Rule. — Multiply £1 by the number of yards ; 55. by the number of quarters ; Is. 3d, by the number of nails ; and add those together for the price of the quantity at £1 per yard ? Take the same part, or parts, &c., of this, as the price is of iSl . Example 1.— What is the price of 97 yards, 3 qrs., 3 nails, at 8s, per yard ? £1 5s. Is. Zd. MuItlpUers 97 3 2 2 )97 17 6 is the price, at £1 per yard. 5)48 18 9 is the price, at IO5. per yard. From this subtract 9 15 9 the price, at 2s. per yard. And the remainder 39 3 is the price, at 8s, (10s.— 2s.) If a yard costs £1, a quarter of a yard must cost 5s. ; and a nail, or the 4th of a yard, will cost the 4th part of 5s. or Is. 3fZ. Example 2.— What is the price of 17 yards, 3 qrs., 2 nails, at £2 5s. 9d. per yard ? £1 5s. Is. Zd. Multipliers 17 3 2 17 17 6 is the price, at £1 per yard 35 15 is the price, at £2 per yard. The price at £l-^ 4=4 9 4i is the price, at 5s. Th© price at 5s. -^ 10=0 8 11| is the price, at 6d. The price at GcZ.-f- 2=0 4 5| is the price, at 3^. And £40 17 9^ is the price, at £2 5s. 9d. EXERCISES. What is the price of 94. 176 yards, 2 qrs., 2 nails, a 15s. per yard > Ans. jei32 9s. 4^d. 95. 37 yards, 3 qrs., at o^l 5s. per yard ? Ans, £47 3s. 9d. 96. 49 yards, 3 qrs., 2 nails, at £1 10s. per yard.? A.ns. £n 16s. 3d. 97. 98 yards, 3 qrs., 1 nail, at £1 15s. per yard.? Ans. £172 18s. d\d. H 226 PRACTICE. 98. 3 yards, 1 qr., at 17s. 6d. per yard.? Am £2 16s. lOid. 99. 4 yards, 2 q^" , 3 nails, at ^1 2s. Ad. per yard ? Am. £5 4s. S^d. 26. Land Pleasure. — Kule. — Multiply £1 by tLe number of acres ; 55. by the number of roods ; and l^d. by the number of perches : — the sum of the products will be the price at £1 per acre. From this find tho price, at the given sum. Example. — What is the rent of 7 acres, 3 roods, 16 perches, at J£3 8s. per acre '? £ s. d. Multipliers 15 1^ 7 3 16 Sum of the products 7 17 0, or the price at JEl per acre. 23 11 the price at £Z per acre. 3 18 6 the price at 10s. per acre. 27 9 6 the price at £3 10s. per aero. Subtract 15 8i the price at 2s. per acre. And 26 13 Si ,j is the price at £3 8s. If one acre costs £1, a quarter of an acre, or one rood, must cost 5s. ; and the 40th part of a quarter, or one perch, must cost the 40th part of 5s. — or Ikd. •'I n EXERCISES. What is the rent of 100. 176 acres, 2 roods, 17 perches, at £5 6s. per acre ? Ans. iE936 Os. 3d. 101. 256 acres, 3 roods, 16 perches, at £6 6s. 6d. per acre .? Ans. ig 1624 lis. 6id. 102. 144 acres, 1 rood, 14 perches, at £5 6s. 8d. per acre ? Ans. £769 16s 103. 344 acres, 3 rbods, 15 perches, at £4 Is. Id. per acre > Ans. £1398 Is. Id. 27. Wine Pleasure. — To find the price of a hogs- head, when the price of a quart is given — Rule. — For each hogshead, reckon as many pounds, and shillings as there arc pence per quart. PRACTICE. 227 Am £2 per yard ? 1 by the and l^d. )ducts will tho price, roods, 16 per aore. r acre. iv acre. t. per aero. V acre. .8s. rood, must erch, must 15 6^. per '6 6s. 6d. s. 8d. per 4 Is. Id. F a hogs- y pounds, Lkkvivve. — What is the price of a hogshead at dd. per quart? A)is. £9 ds. . One hogshead at Id. per quart would bo 63X4, since there are 4 quarts in one gallon, and G3 gallons in one hhd. But G;>x4d.=252<i.=<£l Is.; and, therefore, the price, at dd. per quart, will bo nine times as much — or 9X^1 ls.=£0 9«. EXERCISES. AVhat is the price of 104. 1 hhd. at 18^. per quart ? Ans. £18 185. 105. 1 hhd. at 19<-^. per quart? Ans. £19 195. 106. 1 hhd. at 20d. per quart ? Ans. £21. 107. 1 hhd. at 2s. per quart ? Ans. £25 45. 108. 1 hhd. at 25. 6d. per quart? Aiis. £31 105. When the price of a pint is given, of coxirse we know that of a quarc. 28. Uiven the price of a quart, to find that of a tun— Rule. — Take 4 times as many pounds, and 4 times as many shillings as there are pence per quart. Example. — What is the price of a tun at lid. per quart * £ s. 11 11 4 4G 4 is the price of a tun. Since a tun contains 4 hogsheads, its price must he 4 tiinp,* the price of a hhd. : that is, 4 times as many pounds and shil- lings, as pence per quart [27]. EXERCISES. What is the price of 109. 1 tun, at VJd. per quart? Ans. £79 IGs. 110. 1 tun, at 2Qd. per quart ? Ans. £84. 111. 1 tun, at 25. per quart ? Ans. £100 I65. 112. 1 tun, at 25. 6d. per quart? Ans. £126. 113. 1 tun, at 25 8d. per quart? Ans. £134 85. 29. A nmnher of Articles. — Given the price of 1 article in p(nice, to find that of any number — KuLE. — Divide the number b.y 12, for shillings and ■1 I 11! I< I i 228 PRACTICE. pence; and multiply the quotient by the number of ponce m the price. ExAMPLE.--WIiat is llio price of 438 articles, at 7d caol^ 12)438 365. Gd, the price at Id. each. 20 )255"~6 £12 15 6 the price at 7(/. each. 438 articles at 1<Z. each will cost 438<;.=36s. Gt^. At 7d each 2i¥l5" 6^ '' ''' much-or 7X3G.. Gd.JMs.'SuJ: EXERCISES. What is the price of 114. 176 ib, at 3d. per lb .? Ans. £2 4s. 115. 146 yards, at 9d. per yard ? Ans. £5 9s. 6d ] ;^- ]^^ y^^^,^' '^* 101^. per yard ? Ans. £7 Us. 64 117. 192 yards, at 7id. per yard ? Ans. £6. 118. 240 yards, at 8^d. per yard ? Ans. £S 10s 30. Wages.—mymg the wages per day, to find their amount per year — ° r j> RuLE.--Take so many pounds, half pounds, and 5 pennies sterling, as there are pence per day. Example.— What are the yearly wages, at 5d. per day ? 1 10 5 5 the number of pence per day. 7 12 ] the wages per year. £l^?nfS P^I'^''y;^« equal to 365rf.=240./.+120./.+r,J.= *i-f-lU* +6^/. Therefore any number of pence ner dav mimt be equal to £1 10.. 5d. multiplied by that number ^* What is the amount pei- year, at 119. 3d. per day.? Ans. £4 Us. 3d. 120. Id. per day.? Ans. iglO 12*. 11^. 121. 9d. per day.? Ans. iE13 13*. 9^. 122. 14^. per day.? Ans. £2\ 5*. IQd. 123. 25. 6d. per day .? yl^w. ^'41 I*. 3^. 124 ^d. per day .? Am. j(212 ISs. (]\d. pnACTfc:E. 229 BILLS OF PARCELS. Mr. John Day Dublin, IQth April, 1844. BouKlit of Richard Jones. (/. 15 yards of lino broadcloth, at 13 per yard 24 yanU of tsuperHtu) ditto, at 18 9 !27 yards of yard ^vido ditto, at 8 4 il) yards (;f drugg(it. at . 1 2 yards of sorp;*', at ?)'2 yards of shalloon, at . S 2 10 1 8 £ s. 10 2 22 10 11 5 5 1 14 2 1?> d. () 4 Ans. jCSS 4 10 Mr. James Paul, pair of worsted stockings, at 4 (J pair of silk ditto, at . . ir) 17 pair of tliread dllto, at . 5 2'.) pair of cotton ditto, at . 4 14 pair of yarn ditto, at . 2 15 pair of Women's silk gloves, at 4 I'J yards of Ihinnel, at . . 1 Dublin, mMa\j,\UA. Bought of Thomas Norton, s. d. G per pair „ 4 „ 10 „ 4 2 1}, per yard A,^s. £23 15 4J Mr. James Gorman, 40 ells of dowlas, at ;') 1 ells of diaper, at 31 ells of Holland, at 2'.> yards of Irish cloth, at \~\ yards of muslin, at 13'j' yards of cambric, at s. 1 1 5 10 Dublin, nth May, 1844. Bought of John Walsh & Co G per ell 4^ „ 8 „ 4 per yard 2\ (i [>'t yardsofpi'iMtod calico, at 1 2} 5J 11 Ans. £h4 5 10] 2'30 l'n.\(TlVK mi ■I h Liuly Denny, .'jjimlaofsilk, at . . 12 9 per yard ]■> yards of floAvered do., at 15 G 11-^yardsof Instring, at . 10 J 4 yards of brccado, at . 11 3 12] yards of satin, at . 10 H llg yard,i of velvet, at . 18 Dublin, 20th Maij, 184-1. Bought of Iviuhard Mercer Mr. Jona^i Darling, 5; 15' lb of currants, at 1 7| lb of IMalaga raisins, at l^i* lb of raisins of the sun, at 17 lb of riee, at SI ib of j)Opper, at . . . x k> ;> loa\ OS ol'sugar, weight 32,1 lb. at 81 i:.'. oz. of cloves, at / . " '. D"per"oz Ans. £U 15 10 JDublin, 2lst May, 1844. Bought of William Roper. •s\ d. 4 per lb 0" '6i 1 6 n Aus. £3 13 OJ • Mr. Thomas VVnght, -"""'"' '' "* ^""'' ^'^^^ Bought of Stephen Brown & Co. 252 gallons of prime wliiskey, at 4 per gallon 2r)2 gallons of old malt, at .08 252gallonsof old malt, at . 8 !) Ans. i:204 12 MISCELLANEOUS EXEUCrSES. What is the price of i4 ISa". '2^u. 2. a.>4 lb, at ]\d. per ib .? Ans. £1 ](]s. IQid Jl. 47o6 lb of sugar, at I2}d. per ib .^ Ans. £242 ins. Id. .10 £127 \Qs. 25 pair of silk stoc kings, at C>s. per pair ? A 'tis rUACJ'UK. 231 Ans. i. £242 Am 5. 3751 pair of gloves, at 2^. Cxi. ? Aois. £469 55 6. 3520 pair of gloves, at 3a-. Gd. ? Ans £(516. 7. 7341 cwt., at £2 C)s. per cwt. ? Ans. £16884 Gs. 8. 435 cwt. at £2 7s. per cwt. f Ans. £1022 5s. 9. 4514 cwt., at £2 lis. l},d. per cwt. } Ans. £13005 lO.v. ?,d. 10. 3749f cwt., at £3 15a-. ChI. per cwt. } Am. £14153 17i-. iJJr/. 11. 17 cwt., 1 (jr., 17 lb, at £1 4i-. 9^Z. per cwt. .? £21 lOi'. Sid. 12. 78 cwt., 3 qr.s., 12 lb, at £2 17.?. M. per cwt. .? Ans. £227 lis. 13. 5 oz., G dwt., 17 grs., at 5^. lOrZ. per oz. ? Ans £1 11. S-. 11,/. 14. 4 yards, 2 (|rs., 3 nails, at £l 2.?. 4d. per yard .' A:ns. £5 4s. S^d. 15. 32 acres, 1 rood, 14 perches, at £1 16^. per acre .^ Ans. £5S 4.9. If^Z. 16. 3 ir:dlons, 5 pints, at 7a'. 6^/. per gallon.' Ans, £1 7,v. 21 d. 17. 20 tons, 19 cwt., 3 qrs., 27} ft), at £10 10^ per ton .? Aius. £220 \)s. 1 1^^/. nearly. 18. 219 toas, iij cwt., 3 qrs., at £11 75. Gd. per ton r Ans. £2500 13a-. O-^r/. QUESTIOXS IN PRACTICE. 1. Wluit is practice .' [I]. 2. Vriiy is it so called .' [1]. 3. What is the dilforeneo between ali(|uot, and aliqnant parts.' [2J. 4. Ihm are the ali(|Uot parts of abstract, and of applieato mniibers found } [3]. 5. AVHiat is the difFoiencc between prime, and coni- pouiid aliipiot parts t [3j. fi. [fow is the pri(!e of any denomination fnmd, tli;-it of another beins given ? [(i and .8]. 7. llrtn"- is the -prho of two or more denoi:iination3 found, that of one bein;z uivi-n ? [7 and 9j. S. The p;-icG of ou:.> dc^nominatirM! being given, how do wo find that of any number of another .- [ lOj. I r H 232 PAACTICiC r. ^; F''?.,v '^ ^!''''" ^^ ^''y tlonomination is the aliquot part ot a shilling, how in the price of any number of that tlononuimtion louud ? [llj. 10 AVhen the price of any denoniiuation is the ahquot part of a pound, how is the price of any num- ber ot that denomination found ? [12]. ri3 I ' ^^^'^' ^^ ^"^^"^ ^^ ^^'"^ complement of the price r li. When the complement of the price of any deno- minaum ih the nli^uot part of a pound or shilling, but the price IS not so, how is the price of any number ot that denomination found ? [13 1. 13. When neither the price of a given denomination, 1 or Its complement, is the aliquot part of a pound or M.iUmg, how do we find the price of any number of that denomian'ion ? [14, 15, lb', and 17] . 14 How do we find the price of any number of articles whon t.^e pri(,c of each is an even or odd num- ber ot slullrngs, and less than 20 ? [IS and 19]. lo How is tlio price of a (pumtity, represented by a Tiaxod number, found ? [20] . ^ 1<; How do we find the price of cwt., qrs., and lb, when the price of 1 cwt. is -iven .' [21]. 17. How do we fin.l the price of cwt., qrs., and ib, when th.^pvieoof 1 11, isgiven.^ [22]. 18. h\ vi' is the price of a ton found, when the prico of 1 In IS given ? [23]. ^ 10. How do wo hud the price of oz., dwt., and grs. wh?.i the price 5f an ounce is given } [24]. ' ^20. How do we find the price of yards, qrs., and nails, wucii tlie price of a yard is given .' [25]. 21 How do we lind the price of acre^, roods, and porches .-' [PH]. 22. How may the price of a hhd. or a tun be found, wlion the T;,ri'jo of a <piart is givcm ^ [27 and 28]. ^ 23 Hov; rnny tlie price of any number of articles bo' round, the price of each in pence bciut^ given ? [29] . ^^^" ,^-i*''\'^''« ^^'=^K«« per year found, those - ■ .• day beino given ■' [30] ^ ^ * TARE AND TRET. 233 »ffp m ■15^ TARE AND TRET. 3'. The ^jrross \vei<^ht is tlio wei;;lit both of the goo('h, uii«,l oitho bag, &c., in which thoy arc. Tare Ih an jiHowanco for the bag, &c., wliich contains th(! aiti('h\ Suftle is the wci«;ht which renmins, after deducting the tare. Tret is, uynally, an allowaneo of 4 Vb in every 104 ft), or j'jj- of the weight of goods liable to waste, after the U\V{>. has been d^'dueted. Cioff is an allowance of 2 To in every 3 cwt., after both tare and tret \m\ti been deducted. What reuKiius after makin!? all deductions is called the vd, or nc/il wciu;ht. Diffjrent allowances are made in dilTercnt places, and for different goods ; but the mode of procetMlin;^ is in ^v^l ca^ii'S very simple, and may be understood from the ibllowin<ic — EXERCISES. 1, I>ouo'ht 100 carcasses of beef at 18;$. ikl. per cwt.; gross wciglit 450 cwt., 2 qrs., 23 lb ; tret 8 lb per car- cass. What is to be paid for them ? 100 carcasses. 8 lb per carcass ■ — cwt. qrs. H:) Tret, on the entire, 800 ib=7 IG cwt. qvc. lb. Cross 450 2 23 'J "ret 7 10 443 2 7 at 18s, (Jd. per cwt.=.£410 5s. lOJfL 2. What is the price of 400 raw hides, at 195. 10//. per cwt. ; tlie f^ross weight bein;^ 300 cwt., 3 q'.'s., l.o 11) ; and the tret 4 lb per hide ? Ans. £290 3.7. 2^d.^ 3. If 1 cwt. of butter cost £3, wh:it will be the price of 250 firkins; gro.na weight 127 cwt., 2 qrs., 21 tb ; tare 11 lb per firkhi .? Aiis. £:^(}i) Ss. 0-id. ^ 4. ^Vhat is the price of 8 cwt., 3 (|rs., 11 lb, at Ins. \ fuL per cwt., allowing the usual tret .'^ A its. JLJti 1 Ia\ lO^i/. 234 TAKR AND TRET. G. What U tho price of 8 cwt. 21 lb at lfi» 4i./ por cw . , a I.nvin. the visual fret ? ^L ^7 4/ 81./- (>. Jou.h ^.^ h .J,s. of tallow ; No. 1 wc-ighin. ] J tt qr, 11 11, tare 3 <|r.s, ^u lb ; and No. 2, II cwt nr ' t.u>;';rtect^;t;rtV^^^--^ ^vL^^S;; A. . . cwt. qrs. lb. Orosa M-eight of No. I, K) 1 n (ii-oss weight of No. 2, 11 17 (JrosH weight, . iui-e, Snttlo, Tret 1 lb per cwt. cwt . 21 . 1 2 3 . 1!) . 2 22 19^-§ cwt. qrs. lb. i are 3 20 TuroO 3 14 r~3 G , i. X^5 "tI}'!/. '' ^ '^- The price, at 30,. po, prupo^ti::^'' thirthef tret n.ay be ibund by the following cwt. cwt. qrs. tb. 1 : I'J 2 22 1 lb. 1().'10 7. What is tho price of 4 hhds. of coppnra.s • No 1 WCM,.,,, gross 10 cwt..,2,p..,4 tb,tafo^^;'s. 4lh? 1-^ cwt 1 rp-., tare 3 qrs. I4 lb ; No. 4, 11 cwt 2 'I'-s M I,, tare 3 q.s. IS ib ; the' tret b in" 1 7b ' r <;-t.,-nKl the price 10. per cwt. .^ An! mIoiM. «. Wliat will 2 bags of merchandise corao to • No 1 w,Mg]in,g gross 2 cwt., 3 qrs., 10 lb- No o 'q .J' and at l.v. br/. per lb .? yj^^. £59 2* 8V ^. A merchant has sold 3 bags of pepper • No 1 Weighmo- m-o.SS 3 nwt 9--M.a . V. °o . i'\IW5 ^^^-Ji No. 3, 1 It cwt. 2 qrs. ; No. 2, 4 do th .> cwt., 3 qrs., 21 lb; tare 40 1b •per cwt. ; and the price boin'^ lor/ cwt., 1 qr., 7 lb y come to ? Ans. £74 1.?. 73.3^ por bag; tret per lb. What 1 10. liought 3 packs of wool <F., 12 lb; JN'o. 2,3 weighing. No. 1 , 3 cwt. '-■? qis., ].-) ib ; tare 30 lb cwt., 3 qrs., 7 Ib; No. 3, 3 cwt "'> Rtoue; and at 10.?. 3^ per pack ; tret 8 lb f( amount to ^ per stone. Wliat do tl or f'v.'M-v - ■ --J Je» iAui: ANij via: I. 835 Vo. 1, No. 2, No. 3, owt. qrs. lb. 3 1 12 3 3 7 3 2 15 tb. Tare 30 'J'aro 30 Taro 30 IJross, 10 3 6 'I'aro, 3 6 St. 1 90—3 qrs. G lb. Suttlo, 10 0=70 stones. st. st. lb. lb. 20 : 70 :: 8 : 28 = st. lb. Suttlo, 70 Tret, 1 12 lb. J2 Not weight, 08 4, at lO.-*. Gd. por stono=<£35 16s. 7}^d. 1 1 . Sold 4 packs of wool at 9^. ^Jd. por stono ; woi<^h- ing, No. 1, 3 cwt., 3 qrs., 27 lb. ; No. 2, 3 cwt., 2 qrs., 10 lb. ; No. 3, 4 cwt., 1 qv., 10 lb. ; No. 4, 4 cwt., jjr., G lb : tato 30 lb per pack, ami trot 8 It) for every 20 stono. What is the price > Ans. £49 lbs. 2-^^%d. 12. Bought 6 packs of wool ; weighing, No. 1, 4 cwt., 2 qrs., 15 lb ; No. 2, 4 cwt., 2 qrs. ; No. 3, 3 cwt., 3 qrs., 21 lb ; No. 4, 3 cwt., 3 qrs., 14 lb ; No. 5, 4 cwt., qr., 14 lb : tare 28 lb per pack ; tret K lb for every 20 stone ; and at lis. 6d. per stone. What in the price ? Ans. £11 15s. Sfjfc?. 13. Sold 3 packs of wool ; weighing gross, No. 1, 3 cwt., 1 qr., 27 ft) ; No. 2, 3 cwt., 2 qrs., 16 lb ; No. 3, 4 cwt., qr., 21 lb : tare 29 lb per pack ; tret 8 lb for every 20 stone ; and at lis. Id. per stone. What is the price } Ans. £4\ 13s. l\\\d. 14. Bought 50 casks of butter, weighing gross, 202 cwt., 3 qrs., 14 lb ; tare 20 lb per cwt. What is the net weight } cwt. qrs. lb. Gross weight, 202 3 14 Tare, . .30 qrs. cwt. O I iZ? 14 = i cwt. 202 qrs. o tt). 14 20 4040 lb. 10 5 1 o T 2 of of 25| Net weight, 166 2 161 i of the last, \ =the tare on 3 qr. 14 lb. 2i = Xof thelast, Tare, 4057^ lb = 30 cwt., qr., 2-51 lb. I' 236 TARE AND TRET, lo Ihc gross weight of ton liluls. of tallow is 104 cwt Z qrs., 2b lb ; and the tare 14 lb per cwt. WH-U IS the net weight ? Ans. 91 cwt., 2 qrs., 14-1 lb «J^"/^^ gi-oss weight of six butts of currants is 58 ewt., 1 qr., 8 lb ; and the tare 16 lb per cwt. What is the net weiglit .? Ans. 50 cwt., qr., 1^ ft, 17. What is the net weight of 39 cwt!, 3 qrs., 21 lb • the tare benig 18 lb pe. cwt. ; the tret 4 lb for 101 lb • and the cloff 2 \\, for every 3 cwt. > " ^^ ^^ l il> , cwt. qrs. lb. (xross weight, 39 3 21 Tare, lb. lb. cwt. 18= ! 1^=' ^^- ^ 2=1^8 cwt. qrs. lb. 39 3 21 5 2 23 2 24 1 13 Suttle, . Tret=2'gtli, or Tare, 6 1 13 2 lb m 3 cwt. is the .-^,th part of 3 cwt. i,- u Hence the cloff of 32 cwt. 26 lb is its ^^,th part, or 33 2 1 1 32 2< 2 4 O > Net weight, 32 4 18._ What is the net weiglit of 45 hhds. of toliaeco • weighing gross 224 cwt., 3 qrs., 20 lb ; tare 25 cwt' 3 qrs. ; tret 4 lb per 101 ; cIoiT2 lb for every 3 cwt ? Am. 190 cwt., 1 qr., 14^^ lb. ^ 19 What is the net "weight of 7 hhds. of sn^-nr weighing gross, 47 cwt., 2 qrs., 4 lb ; tare in the whol..' 10 cwt., 2 qrs., 14 lb ; and tret 4 ib per 104 \h > An^ 3o cwt., 1 qr., 27 lb. ^ ' ' 20. In 17 cwt., qr.. 17 lb, gross weight of V^alls how much net ; allowing 18 lb per cwt. ta?e ; 4 lb per' 104 lb tret ; and 2 \h per 3 cwt. cloff? Ans. 13 cwt., 6 qrs., 1 lb nearly. ' QUESTIONS. 1. What is the gross weio-ht ? fSll 2. What is tare? [31]. ^ ^ "'' 3. What is suttle .? [31 ] . 4 What is tret.? [311 ' 5. What is cloff.? [31]. 6. "Wliat is the iit-t weight.? [31]. 7. Are the allowances made, always the same ? [31], qrs lb. 3 21 1 13 o i> Ji 1 4 20 O > Am. 237 . i; SECTION VII. INTEREST, &c. 1. Interest is the price which is allowed for the nso of money ; it depends on the plenty or scarcity of the latter, and the risk which is run in lending it. Interest is either simple or compound. It is simf^h when the interest due is not added to the sum lent, ^^, as to bear interest. It is compound when, after certain periods, it; is made to bear interest— being added to the sum, and considered as a part of it. The money lent is called the principal. The sura allowed for each hundred pounds " per annum" (for a year) is called the " rate per cent." — (per iilOO.) The amount ia the sum of the principal and the interest due. SIMPLE INTEREST. 2. To find the interest, at any rate per cent., on any Bum, for one year — lluLE I. — Multiply the sum by the rate per cent., and divide the product by 100. Example.— What is the interest of £072 14s. U. for cno year, at 6 per cent. (XG fur every £100.) £> s. (J. 672 14 3 40-30 20 5 G 7-25 The quotient, £40 7s. 3f/., is the iLtereet required. 'J.Ort J lii We h ave divided by 100, by merely altering the decimal point [Se<3. I. 34J O' kS IN CERKST. II till- Interest w'.'i'c 1 percent,., it woulrl be the Imiidredth part of the principal— or the pj-ineip;il multiplied l)y ^^-f, ; but being b per cent., it is t> times iid much— or tlie principal mul- tiplied by y|^. 3. Rule II. — Divide the interest into parts of cClOO; and take corresponding parts of the principal. EsAMPLK.— What is the interest of £32 4s. 2d., at G per cent. '. ^ £G = £5+£l,ov£^^lAns£^^-^5. Therefore tho in- terest is the J^ of the principal, plus the I of the J-. £ 20)32 s. d. 4 2 5) 1 12 2r} is the interest, at 5 per cent. 6 5]- is the interest, at I per cent. And 1 18 7f is the interest, at 6 (5+1) per cent. '* EXERCISES. 1. What is the interest of ^£344 lis. Qd. for one year, at per cent. > Ans. £20 1 3*-. 1 0\d. 2. What is the interest of .£600 for one year, at 5 per cent. ? Am. i230. 3. What is the interest of dE480 15.y. for one year, at 7 per cent, t Jlns. ii33 135. 0|rZ. 4. What is the interest of ^£240 10s. for one year, at 4 per cent. > Am. £9 12s. Md. _ 4. To find the interest when the rate per cent, con- sists of more than one denomination — ^ lluLE. — Find the interest at the higliest denomina- tion ; and take parts of tliis, for those which are lower. The sum of the results will be the interest, at the given rate. Example. —What is the interest of £97 8s. 4d., f(y one year, at £o lO.s. per annum '? £5 = £yi,0; and 10.'. = £,5,. £ s. d. '^ 20)97 8 4 10)4 17 5 is tho interest, at 5 per cent. 9 9 is the interest, at lOs. per cent. And 5 7 2 is the interest, at £5-{-10s. per cent. £5 INTEKEST. 239 At 5 per cent, tho intevcst is the .1,- of fhe piMneipal ; at lOs. per c.-nt. it is the j\ of wluit it U at 5 |.er cent. There- fore, at £5 lOi'. per cent., it is the sum of huth. 5. What is tho interest of /J371 IDs. ly. for one year, at i:'3 155. per cent. ? Am. jSIS 18.v."llf^. ei. AVhat is the interest of i^84 ll.y. lOirZ. for one year, at £4 5.v. per cent. } Ans. j£;3 II.9. lOfr/. 7. What is the interest of JCOI O5. 3|<Z. for one year, at £6 12.9. 9^/. per cent. } Am. M O5. \0\d. 8. What is the interest of £ms bs. for one year, at £b 14a-. i)d. per cent. } Am. £bb 8a-. 8r/. 5. To find the interest of any sum, for several years — Eui.E. — Multiply the interest of one year by the num- ber of years. Example.— AVJiat is the interest of £32 145. 2(/. for 7 years, at 5 per cent. '? £ A-. d. 20)32_14_2_ i 12 81^ is the interest for one year, at 5 per cent. And 1.1 S 11^ is the interest for 7 years, at 5 per cent. This rule requires no explanation. EXEilCISKS. 9. Vvniat is the interest of £U 2s. for 3 years, at (3 per cent, r Ans. £2 U)s. dd. 10. What is the interest of ,£72 for 13 years, at m 10a-. per cent. } Ans. £m IG.v. 9*^/. 11. What is tho interest of £Sb3 Qs. i)\d. for 11 years, at £4: V2s. per cent. } Ans. £431 vSs. l^d. ().. To find the interest of a given sum for years, niontlis, &c. — lluu:. — Having found the interest for the years, as ah-eady directed [-2, &c.], take parts of the interest "jr that of tho mouths, &( the results. and tl'sn add <U ! I '■rt. l^ui I j-,jj i Hi ^•40 INTEREST Example —What is the iz^tercs,* of J£86 85. M. for 7 vcars and 5 months, at 5 per cont. > ^ 20)86 8 4 4 G 5 13 the interest b> : T?a>, ai 5 jxv • cep 1 « o.~^"" A S *if ?^ *^'® interest for 4 monthv 1 8 9^-^4 =0 7 21 IS the interest for 1 month. And 32 llj is the required interest. EXERCISES. 12. What is the interest of ^£211 5^. for 1 year and 6 months, at 6 per cent. .? yl-^w. ^19 0.9. 3d 13 AVhat is the interest of £514 for 1 year and 7-i months, at 8 per cent. > Ans. £66 16^ 4U ' 1-1. What is the interest of £1090 for l' year and 5 months, at 6 per cont. } Ans. £92 I3s 1;^ What is the interest of £175 lO.^. 6^. for 1 year and J monriis, at 6 per cent. .? Ans. £16 135. 5//^.^ o. A\Iuit IS the interest of £571 15.. for 4 years and 8 months, at 6 per cent. .? Ans. £160 1.. 9-^-./ 17 VViiat IS the interest of £500 for 2 years and 10 iiiwntlis, at 7 per cent. ? Ans. £99 3.y. Ad lb". What is the interest of £93 17.':. Ad. for 7 yc^a-s ,}^T'^^'f' ^'^' ^ P^^' «^"t- • ^'I'i'^-. ^14 11.. lid " and ^ ^ ;f '•' /'-' ^"^''''^ '^^ ^'^-^ '^-^- ^'^^- f^^- 8 y^^^ and S mouths, at o per cent. > Ans. £36 ll.v. 111,/. O) or b, &c., per cent. ' At 5 per cent. — * ■ KuLE.— Consider the years as sh:lHn..s, and the montlis as pence ; and find what ali-pot pa^ or part ^fS^^^n^r^^"^- ^^-*^^-^--4^tor^a.. To find the interest at 6 per cent., find the interest at 5 per cent., mid to it add its fifth part, &c. Ihe mtcrest at 4 per cent, will bo the per cent minus its fifth part, &c. interest at INTKRE6T. 2^1 d. for 7 years 4^ 5 p<\ • cep s. bh. 1 year and ear and 7^^ yx^ar and 5 for 1 year ''• ^1 0*^- 3r 4 years s. 9^(1, ars and 10 ■or 7 years Is. l\d. n- 8 years v. 111^/. y time, at and tliG t or parts t or parts LJ interest iterest a^ ^ 8. Example L— What is the interest of X427 5^. 9c/. for years jind 4 luuiiths, at 5 per cent. '\ years and 4 months are represented by 6s. 4r/. : but 0.. 4,/.=...s..+],.+W.==<-}-J^ of a pou)td + the i of ths J,. 4)427_ .5__0_ 5)100 To ,5y is tlie I of principal. ;])21 7 31 is the ^V(l of]) of principal. _-!-._r_A ''^ ^^'"^ s'" <^'^ ''^' ^'') ^''" Pi'i'^^ip'd. And l;>.5 G 1^ is the required interest. The intcres^t of £1 for 1 year, at 5 per cent., would be 1.. lor 1 inouth 1</. ; for any number of years, the same nuinl)or ot shillnigs; for any number of nioiitlis, the same number of pence ; and for years and months, a corresponding nund>er of shiUmgs and pence. But whatever part, or parts, these sliil- xHigs and i>ence are of a pound, the interest of any other sum tor tJie sauio time and rate, must be the same part or parts of tJiat otlier sum-since the interest of any sum is proportional to the interest of JSI. KxAMPi-E 2.— AVhat is the interest of £14 2s. 2d. for G years and 8 months, at G per cent. ? G.'v. 8^/. is the .' of a pound. £ s. d.' 3)14_2^2_ 5)4 14 {)} is tlio interest, at 5 per cent. 18 W'l is the interest, at 1 per cent. 5 12 KJi is the interest, at G (5-f-l) per cent. EXERCISES. 20. Find the interest of .£1090 17.?. Qd. for 1 year and 8 months, at 5 per cent. .? Am. £90 18.v. \\d. 21. Find the interest of £976 14,?. Id. for 2%^ears and 6 months, at 5 per cent. } Am. £122 \s. 9|rZ. 22. Find the interest of £780 17s. fi^i. for 3 years and 4 iiiontlis, ut G per cent. > Am. £1.^)6 'M. 6d. 23. What is tlie interest of £197 lis. for 2 years and 6 montli.s, at 5 per cent. ? Am\ £24 13.?. l{)ld. 24. What is the interest of £279 lis. for 74- months, at 4 per cent. .- A as. £6 19s. 9,-^,//. 7U( Jit; 2.1. What IS iho. iuterest of £790 IGs. for 6 and S months, at 5 per cent. > A'us. £263 12j. year -^m y 212 INTKREST, 26 What is the interest of ^^121 2s. \U. for 3 years and .3 inontiis, at 5 per cent. ? Ans. £oq 3^ 53,^ "^ 27 ^VliMt i,s the interest of i2l837 4*. 2d. foVs'vca.-a and K) inontlis, at 8 per cent. } Am. £563 S.y. 3d. 9. When the rnU^ or nuin])er of years, or loth of thein, are expressed by a mixed nuinber- l.ULE.— Find the interest for 1 year, at 1 per cent , and multiply this by the number of pounds and the frac- tjun of a pound (if there is one) per cent. ; the Mini uf these products, or one of them, if there is but one, will give the interest for one year. Multiply this bv' the number of years, and by the fraction of i year (if^lhe e f one) ;_ and the suni of tliese products, or one of them' If there is but one, will be the required interest atfpoTce'atT*"^ *'' interest of i:21 2.. 0./. fur 3| years £21 2s-. G./.^100=4.. 2-ld. Therefore .t s. d. 4 25 is the interest fori year, at 1 per cent. lit Jo. 1 1 1' is the interest fori year, at 5 per cent. a 1 ? 1 A n^ the interest for 3 years, at do. i> lo luj IS the interest for 2 ofa year (£1 Is. l^^^.x"'),; 3 19 31 is the interest for 3^- years, at do. J^^':.^''^ '' *^-i'^terestof £300fbr5.years, £300^100=3 LUhe interest fori year, at 1 per cent. 9 OLs the interest fori year, at 3 per cont. __ ___^^ tlie interest for 1 year, at £■' (cC3x^) 11 5 is the interest fori year, at 3| per cent. 56 5 Is the interest for 5 years, at ^ por cent - 10 3 IS tlio do. for l- pru- (i:5 12^. G^r/.J-2A And 04 13 9 is the interest I) r^■■t or ir^ years, at Z} do. 'or 3 ycarfl , 6^1. fur 3 ycai'3 <s. 3d. •1* both oC por cent., (1 the fruc- he sum vi' t ono, Avill lis by the ' (if (lierc B of them, !t. >r 3| years x;'),utJo. 5^ years, per cent, oer cont. por cent. ' peroenl t 3^ do. rXTERRST. EXRRCISKS. 243 28. What is tliG interest of JE379 2s. Gd. for 41- years, at 5 1 por cent. ? Aiis. £<ji 5.5. 5^/, ^ ^ » 29. What is the interest of .£640 \Qs. Qd for 24 years, at^4| per cent. > Am. £72 1,?. 2j\d. 30. "What is the interest of £600 IO.9. Qd. for 3i years, at 5|- per cent. } Ans. £11.5 2^. QJ-fZ. '' 31. What is the interest of £21? ^s^.\id for e^ years, at 5 5 per cent. .? Ans. £81 8,?. bl-d. "* 10. To find the interest for days^ at 5 per cent — '/^.^^•— ^^"Itiply the principafby the number of days, and divide the product by 7300. E.VAMPLK.— Whatis the interst of i;2G 4.s. 2d. for 8 days? £> s. d. 2G 4 2 8 201) 13 4 20 4193 12 r300)50320(6:!-|-'W. •43800 ' 6520 U ItVZT'"'] "l^T'*/.' .^'^« ^' ^' 7^— «ince tlie remainder IS gieatey than lialf the divisor. The interest of £1 for 1 year is £J^, and for 1 day Jj-f-3G5= 20K3tl5="300; that is, the 7300th part of the principal. Therefore the iritorest of auy other sum for one day, is the .?00th part of that sum; and for any number of da^s. it a that number, multiplie.l by the TSOOtli* part of tl>e princ ipal- nunrbpl'of T "'' 'T^- '-^'/"f' '^'' ^'^""^^^^ multiplied bf he number of days, and divided by 7300. ^ EXKRCISES. 33. Find the interest of £140 lO.f. for 76 days, at 5 per cent. Ans. £1 9s. 3^%^jd. 33. Find the interest of £300 for 91 days, at 5 per cent. '•'■'• ^"^ '-'- no., 7 -^ ' 1^^ Ans. £3 I4s. Q^d 34. What is the interest of £800 fur 61 day i ; i ,i' , ^ 1 per cent. ? Am. £6 1.3.s-. S%%d s, at b 241 I.NTKKKST. 11. To fmr] t],o intovost f<n- dnys, at .^;/7/ other rate— pitrls of llii KXAMI'MO fliiys, at cCG X3324 Gs. £ .V. 5)5 2)1 OJU And G lo" je 1+105. This rule 1 111,1 the mtercj.st at 5 per cent., and take ■* tor the remainder. —What is the interest of Xr.324 C.<. 2<Z. for 11 ivs. per cent. ? 2^.Xll-^7;500=£5 0.. 2],/. Therefore 2| is the interest for 11 days, at 5 per cent. J .J IS the interest for 11 days, at 1 per cent. JJ_ IS the interest for 11 days, at 10,v. per cent. 2J is the interest for 11 days, at jGG 10*-. (£5-f requires no explanation. EXERCISRS. w ?f' ^u'^V" *^»e interest of £200 from the 7th May to .the 2bth September, at 8 per cent. } Ans. £Q As 30. mat is the interest of ^£150 15^. Qd. for 53 aays, at 7 per cent. } Am. £\ \0s. 7^d 37. What is the interest of d8371 for l"year and 213 days, at 6 per cent. .? Ans. i235 os. Qd A.!ll' .Tl''''^ "' ^\ ''l^'^'f^ °^ ^'-^^ ^'''' ^ y^-^'' and 135 aays, at 7 per cent. } Ans. £23 0.s\ 3^^;?. Sometimes the number of days is"the\liquot part of a year ; m which case the process is rendered more easy l^^^-^:ti^^ '^''''' '' ^^'^ '- ' ^- -^ 1 year and 7-3 days=] i year. Hence tlie required intercsit il75 '^r7'' ^'" 1 /f- +i.t« lifth part. ]^ut Ihe interest':? fjiio tor 1 year, at the given rate is £14. Tlierefore its £1g'ig1 gi^«^ti>^« i« X14+i:y=:£i4+i2Ta..i 12. To find the interest for mon/Jis, at 6 per cent-^ ±IULE.— if the number expressing tlie months is even, nmtoply the prmcipal by half i/>f number of month and dn,de by 100 But if it is odd, multiply by tho hali of om Jess than the number of montlis ; divide the result by 100 i and add to the quotient what will bo obtained if we divide it by one les than the nn.mIo of 11 INTEUIiST 345 ! ti KxAMiT.K 1.— What is the interest of X72 (Js. id. for 8 months, iit G per cent? £ ,s. (I. 72 4 4 £2H0 5 4 20 17-85s. The required interest h £2 Us. lOUl. 12 10-24r/. 4 0-90= J f/. nearly. Solving the question Ity tiic rule of three, wo shall have — ;eiOO : i:72 Gs. 4^/. : : JCG : £72 6.s-. 4f/.x8xG 12 : 8 l(Xrxl2 =(^^ivul. ing Itoth numerator and denoniinator by G [Sec. IV. -' X72G.9. 4r/.x8xG-^G .£72 Gs. 4r/.xS 100 x 12-^0 = lUO X 2 ~" = (dividing ooth numerator and donominator by 2) ' ' -" ■^^^yczo_J_o'~~ £12 G.;. Aily^A 100" — that is, the required intore.sfc is equal to tlic given sum, nuiKiplied by half the number which expresses the montliH, and divided "by 100. Ex-AMPi.E 2.— AVhat is the interest of £84 Gs. 2d. for 11 months, at G per cent. ? 11=10+1 10-^2=5. £ s. d. 84 6 2 •) One loss than the given number of n-iHol^ inonths=10. 20 £ s. d. 4oOs. i0)4 4 Ojj- is the interest for 10 niontliii, at U ptT oc'iit. 12 8 5| is the intore!.t lor 1 nujutl), ut Siinie rattj. 3-70(7. And 4 12 9 is the iutciest lor IJ (ID-j-I) moutU-i, ut ti ;•* 4 2-80f.=.^J. nearly. Tlie interest 11 -1 month, pUis the "udovost of II — I monlh -r-1 1 — 1- 11 months is evidently the interest ivf I .M 948 INTEItnsT. ual > 'IXKKCISES. 39 What k tlio intorost of £250 17s 6d for <( «fr ,;,,.'"■ "!,"""■"■'"'''' ■^■5^1 15s. for 8 months. «*'. per (,..(.- ^«. Jtaa 17s. 4.id ' 6 pt or„t.".' "i!;.: i^'^rres"' ^'''° '"' ' "■"'"■'^' ^' 6 p1fr ofntt ^f i"r^' "' ■'"'' '" '" '™"*^' "' at P,i^ "^^^r^^^^t ''■ '- ' -""■»' ...i' v^ir^*^ *''? '"''"'™' '« ?»■<! l>y -^nj/s, multinlv the sam by the nnmboi- of days which have elamcd b^fom any payment wa. made. ' Subtract Z trjfavu „, and multiply the remainder by the number of S wh.eh passed between the first and second paymon?, Ir r ^"'" "'T'^ P^>'™""' »■«' '""'"-ply «l.is nS: seconJ !nd "ir^'' "^ ''"^^ '^'™'' passL/ between the days more £20 : in 15 mor^ £^o "^ i f .f'*'"^^ ~^ '^ • ^" '^ day,, an, the ^SeTat"' Cf Jlrj^^e^'ireit X days. £ day. nZx 6= 702xi 1 lOOx 7= 700x1 i^0xl5=1200xlf=°C^^^<^- 48x60=3168x1 J Wo?e''''* '^'^"' ^^^ 1 ^-y- -*S Ver cent., is 15.. 9|c/ INTERRST. 24/ ..... i fi)0 15 \)'^ is tho iutorost, at 5 pei cent. '^ 2^ is tho interest, at I per cent. Jl)0 18 11^' i8 tho intorest, at per cont. G 4 is tho interest, at 2 per cent. And 1 5 3J is tho interest, at 8 per cent., for th*^ given sums and iiiuos. If tho entire sum were G days unpaid, tho interest would be tho aafoe as that of 6 times as much, fbr 1 day. Next, £100 due for 7 days, sliuuM proiluce as much as £700, for 1 day, &c. And all tlio sums du ■ for tho diffcrci>t periods should produce as much as the sum of their equivalents, in 1 day. EXERCISES. 45. A merchant borrows i2250 at 8 per cent, for 2 years, with condition to pay before that time as much of the principal as he pleases. At tho expiration of 9 months he pays jL'80, and 6 months after £70 — leaving the remainder for the entire terra of 2 years. How much interest and principal has he to pay, at the end of that time > ins. J£;i27 16^. 46. I borrow ^£300 at 6 per cent, for 18 months, with condition to pay as much of the principal before the time as T please. In 3 months I pay £G0 ; 4 months after i^lOO j and a months after that £75. How much principal and interest am I to pay, ut tho end of 18 montiis.? Ans. £,"^9 los. 47. A gives to B at interest on the 1st November, 1804, £6000, at 4^ per cont. B is to repay him with interest, at the expiration of 2 years —having liberty to pay before that time as nmch of the principal as he pleases. Now B pays £ 900 The IGth December, 1804, The 11th March, 1805, The P.i'Ui March, The 17th August, The 12th February, 1806, 1260v 600 800 1048 How much principal and interest is he to pay on th# 1st Novembor, 1806 ? Ans. £1642 9^. 2if ||-fZ. 48. Ticntat interest £600 the 13th May," 1833, for 24n INTIRIST. 1 year, at 5 per cent—with condition that the receiver may diHchargo as much of the principal before the tinm ns ho pk.as,vs. Now ho pays tlio i)tli July .£^200 ; and the 1/th beptember i^l50. How nmch principal and mtorost IS ho to pay at the expiration of the year ? Ans. £26Q 13s. b^\d. ^ I ^'^' }^}^yv^^ that the pupil, from what he hag learned of the properties of proportion, will easily un- derstand the modes in which the following rules are proved to be correct. Of the principal, amount, time, and rate— given any three, to find the fourth. ^ ^ Given the amount, rate of interest, and time : to find the prmcipal — ' ^ Rule.— Say as £\Q0^ plus the interest of it, for the given tune, and at the given rate, is to ^100 ; so ia the given amount to the principal sought. ^^ExAMPLE.-Whiit will produce £862 in 8 years, at 5 per giv1n'rfe.^'Tf2ir? "'""^ '""^ ^''' ^" ' ^^^ ^' *^« £140 : £100 : : £862 : J^^^^ =£615 14.. 31^. When the time and rate are given n.„^ll!? • ^^^ °^^''"' ^"""^ • ' ^"*®^^'** of £100 : interest of that oilier sum. By alteration [Sec. V. 20], this bocomes- th^^mn" '°*"''^* ""^ ^^^^^ '' ■ "^"^ °*^®'' '"'"' • ^°*^''^^*^ ^^ rw v%T-i"° "th« first + the second : the second," &c. L^ec. V. 2\)] we have — f^fi^^ i I*' interest : £100 : : any other sum -f- its in- terest . that sum— which is exactly the rule. EXERCISES. 49. What principal put to interest for 5 years will nmount to £402 10.., at 3 per cent, per annum ? Ans, 50. What principal put to interest for 9 years, at 4 percent., will amount to £734 S.. .=^ Ans. INTEREST. 249 51. Tho amount of a certain principal, bearing inter- est for 7 y(;ars, at 5 per cent., u jL'334 16*. What is tho j)riucipul ? Ans. £24ii. 1.'5. Given tho time, rato of interest, and principal— to find tho amount — lluLK.—Say, as JEIOO is to JCIOO plus its interest for the given time, and at the given rato, so is tho given sum to tho amount required. ExAMPLK.— What will £272 jomo to, in 5 years, at 5 per C6Ht* f ^'fi}^r (='^1^0-f-£5x5) is the principal and interest of *.10U fur 5 years ; then — £100 : £125 : : £272 : ^~=.£ZiO, the required amount. We found by tho last rule that £iOO+its interest : £100 : : any other sumf its interest : that sum. Inversion [See. V. 20] changes lliis into, £100 : £100-f-its interest : : any other sum : that other Bum-fits mterest— which is the pi.vBont rule. EXERCISES. 52. What will £350 amonnt to, in 5 years, at 3 per cent, per annum ? Ans. £402 10a\ 53. What will £540 amount to, in 9 years, at 4 per cent, per annum ? Ans. £734 8.y. 54. What will £248 amount to, in 7 years, at 5 per cent, per annum ? Ans. £334 16s. 55. What will £973 4s. 2d. amount to, in 4 years and 8 months, at 6 per cent. ? Ans. £1245 145. l^d. 56. What will £42 3^. Qid. amount to, in 5 years and 3 mouths, at 7 per cent. ? Ans. £57 13.?. lOirZ. 16. Given the amount, principal, and rato — to find the time — Rule. — Say, as tho interest of tho given sum for 1 year is to th • • • - juired time. given interest, so is 1 year to the re- 250 INTEREST. X14 l5. 8rf. (the interest of £281 13s. 4d. for 1 year r21) • £56 6s Sd. .^56 Gs. iM. (the given interest) lequircd number of years. 1: X14 Is. 8(/.=^' *^« 17. iience bij • + . r , .'^^'%> to find the time— Divide the InZlt '^l^'"^'" ^"'^'^^^^ ^^^ 1 3^^^^' i«to the entire interest, *ud the quotient will be the time. in/ln'f •'^'^^''*' *^'" P"°«ipal, and rate beina; c-iven, the ntere«t is prcpo-uonal to the time; the longer the Jme the more the interest, ^ud the reverse. That is- ' ihe interest for one time : the interest for another • • the former time : the latter. <iuouier . . ^ Hence the interest of the given sum for one year rthe nterest for o,ic time) : the given interest (the interest of the same sum for amther time^ • • 1 vpo^ h\Z ,. ^^^^^f.^* produeedthe former) : thHfi- sought tiatXiX^^^^^^^ uuced the latter)-.which is the rule. ^ EXERCISES. 57. la what time wciild .^300 amount to ^£372, at 6 per cent. > Ans. 4 years, ' 58 In what time would £211 5s. amount to ^£230 moniht' ^'' ''''*•• ^^'- ^" ^ y^^^ ^^^ 6 59. When would £561 15s. become £719 Os 95^ at 6 per cent. ? Ans. In 4 years and 8 months. * ' 60. When would £500 become £599 3s. 4d., at 7 per ""^^V .,r<^'""- ^^ ^ y^^^"« ^n^ 10 months. ^ 61. When will £436 9s. 4d. become £571 8s Ud at 7 per cent. ? Ans. In 4 years and 5 months. ' ' ' the rat^""' *^'^ amount, principal, and time— to find EuLE— Say, as the principal is to £100, so is the given interest to the interest of £100— which will give he interest of £100 at the same rate, and for the same he rat ' ' ^ *"''''' ^""^ *^^' ^''^*^'^* ^" ^« INTEREST. 251 Example.— At what rate will £350 amount to £402 10s in 5 years ? £350 : £100 : : £52 10s. : ^^^ 10.-. x 100 350 =£15, the in terest of £100 for the same time, and at the same rate IJien 'j=3, is tbi required number of years. We have seen [14] that the time and rate being uie same, £100 : any other sum : : the interest of £100 : interest of the other sum. This becomes, by inversion [Sec. V. 29] — Any sum : £100 : : interest of the former : interest of 100 (for same number of years) . But the interest of £100 divided by the number of years wliich produced it, gives the interest of £100 for 1 year— or, in other words, tlie rate. EXERCISES. 62. At what rato will c£300 amount in 4 years to i£372 ? Ans. tj per cent. 63. At what rate will £248 amount in 7 years to £334 16s. ? Ans. 5 per cent. 64. At what rate will £976 145. 7d. amount in 2 years and 6 months to £1098 IQs. 4^d. } Ans. 5 per cent. Deducting the 5th part of*the interest, will give the in- terest of £070 145. Id. for 2 years. 65. At what rate will £780 175. Gd. become £937 Is. in 3 year.1 and 4 months ? Ans. 6 per cent. 66. At what rate will £843 5^. 9d. become £1047 Is. 7|<-7., in 4 years and 10 months ? Ans. At 5 per cent. 67. ^ At what rate will £43 25. 4JyZ. become £00 75 4J-rf., in 6 years and 8 months } Ans. At 6 per cent. 68. At what rate will £473 become £900 135. fii^Z in 12 years and 11 months ? Ans. At 7 per cent. COMPOUND INTEREST. 19. Given the principal, rate, and. time — to find the amount and interest — lluLE I. — Find the interest due at the first time of payment, and add it to the principal. Find the interest ■t H r i 1 1 • t 1 , 1 1 I ■ m 1 ■■ ;; i ■ 1 1; , K i. :. J . ; , ilHiiJ ■i m m 252 INTEREST. of that sum, consiJorotl as a new priucipal, and add it to what it would produce at the next payment. Con- sider that new sum as a principal, itud proceed as before. Continue this pi-ocess through all the times of payment. Example.— What is the compound interest of £97, for 4 years, at 4 per cent, lialf-yoarly '? £ s. d. 97 ^ 3 17 7.i is the interest, at the end of 1st half year. 100 17 1\ is the amount, at end of Isfc half-year. 4 8^- is the interest, at the end of 1st year. 104 18 3'/ is the amount, at the end of 1st year. 4 3 11| is the interest, at the end of 3rd half- year. 109 4 2 7 3 is the amount, at the end of 3rd half-year. 3L is the interest, at the end of 2nd year. 113 4 118 4 14 9 0.} is the amount, at the end of 2nd year. 10 9.^ is the interest, at the end of 5th half-vcar. 4 is the amount, at the end of 5tli luilf-year. 5 is the interest, at the end of 3rd year. 122 14 9 is the amount, at the end of 3rd year. 4 18 2\ is the interest,* at the end of 7th half-year, 127 12 Hi is the amount, at the end of 7th half-year, 5 2 1^ is the interest, at the end of 4th year. 132 15 0;i is the amount, at the etui of ith year. 97 is the principal. And 35 15 0^- is the compound interest of £97, in 4 years. 20. This is a tedious mode of proceeding, particularly when the times of payment are numerous ; it is, tlun-e- foro, better to use the following rules, which will be found to produce the same result — KuLE II. — Find the interest of £1 for one of the payments at the given rate. Find the product of so luany factors (cdch of them c-Gl-fits interest for o!ie payment) as there arc times of payment ; multiply this product by the given principal ; and the result v/ill bo the principal, plus its compound interest for the given Ur INTEREST. 253 time. From this subtract the principal, and the remain- der will be its compound interest. Example 1. — What is the compound interest of £237 for 3 years, at 6 per cent. '? £0Q is the interest of £1 for 1 year, at the given rate ; and there are 3 payments. Therefore £1-06 (^£l-\-£0-\j) ig to be taken 3 times to form a product. Hence lOGxlOOx l-06x£237 is the amount at the end of three years; and l-0Gxl06xl-06x£237— £237 is the compound interest. The following is the process in full — £ 1-06 the amount of £1, in one year. 1'06 the multiplier. 11236 the |j,mount of £1, in two years lOG the multiplier. 1-191016 the amount of £1, in three years Multiplying by 237, the principal, £ s. d. wo find that 282-270792=282 5 5 is the amount • and subtracting 237 0, the principal, we obtain 45 5 5 as the compound interest>. Example 2. — What are the amount and compound inte- rest of £79 for 6 years, at 5 per cent. 1 The amount of £1 for 1 year, at this rate would be £10.5. ^ Therefore £1-05 X 105 X 1-05 X 1 '05 X 105 x 1-05x79 is the amount. &c. And the process in full will be — £ 1-05 105 11025 the amount of £1, in two years. 11025 1-1^1551 the amount of £1, in four years. 1-1025 1-34010 the amount of £1, in six years. £ s. a. £105-86790=105 17 4| is the required amount. 79 And 26 17 4| is the rcqnired interest M 2 2.11 tNTEKKST Example o. — Whataro the aiTJount, and compound interest of £27, for 4 years, at £2 10s. per cent. Iialf-yearly. The anionnt of XI for ono pnyinont is X102:>. Therpfore ClO-iry X I- 025 X 1 025 X 102o x 1025 x 1-025 x 1025 x I 025 X27 is the amount, &o. And the process in full will be £ 1025 1025 1 05003 the amount of XL in one year, 105003 "^ T- 10382 the amount of XI, in two years. 1-10382 ■ 1-21842 the amount of XI, in four years. 27 , ^ £ s. (17 X32-8U734=32 17 11| is the required amount. : 27 And 5 17 li| is the required interest. 21. Rule IIT.— Find by the interest table (at the end of the treatise) the amount oi £,\ at the .f^iven rate, and for the given number of payments ; multiply this by tho given principal, and the product will be the required amount. Prom this product subtract the principal, and tho remaiader will be the required compound interest. Example.— AVhat is the amount and compound interest of X47 lOi. for 6 years, at 3 per cent., half-yearly '? < X47 10,y.=X47-5. We find by the table that X1-4257G is the amount of XI, for the given time and rate. 47-5 is the ■ .altiplicr. ~rr~~ ^ •"'• ''• 67-7230=67 14 5'' is the required amount. 47 10 And 20 4 5^ is tho required interest. 22. Tlule r. requires no explanation. li,KAso\' OK F.uLK II. — Whon the time and rate are Mie Bfvme, ivfo priicipnls are proportional to their corresponding amounts. Thcrofore £1 (one principal) : £1 03 (its corresponding amount) : £10G (.another principal) : £1-00 X lOG (its corresponding amount). IXTF.UKST. 2i?5 Ileneo the amount of £1 for two ycirs, is £106xl'06— or the product of two factors, each of them the amount of £1 for one yeur. Again, for similar reasons, £1 : £1-06 :: £l-06Xl-0G : £1 -OGXl-OGXl-OG- Hence the amount of £1 for three years, la £l'06xl'06Xl*06— or the product of three factors, each of them the amount of £1 for one year. The same reasoning would answer for any number of pa^'- ments. The amount of any principal will be as much greater than the amount of £1, at the same rate, and for the same time, as the principal itself is greater than £1. Hence we multiply the amount of £1, by the given principal. Rule III. requires no explanation. 23, When the decimals bejonie numerous, we may proceed as already directed [Sec. II. 58]. We may also shorten the process, in many cases, if we remcTaber that the product of two of the factors multiplied by itself, is equal to the product of four of them ; that the product of four multiplied by the pro- duct of two is equal to the product of six ; and that the product of four multiplied by tho product of four, is equal to the product of eight, &c. Q'hus, in example 2, M025 (=l-05xl-05) xi-1025=105xl-05xl-05xl 05. EXERCISES. 1 . What arc the amount and compound interest of £91 for 7 years, at 5 per cent, per annura ^ Ans. £12S 05. l]d. is the amount; and .£37 05. lit/., the com- pound interest. 2. What are the amount and compound interest of £142 for 8 years, at 3 per cent, half-yearly.? Ami. £227 175. 4ld, is the amount ; and £85 175. 4^^., the compound interest. 3. What are the amount and compound interest of £63 55. fi , ars, at 4 per cent, per annum ? Ans. £90 05. 5f//.'is the amount; and £26 155. Sfc/., tho compound ixiterest. 4. What are the amount and compound interest of £44 05. dd. for 1 1 years, at 6 per cent, per annum .'* f\' 256 INTEREST. Ans. £84 Is. tul. is the amount; aud £39 155. Sd the compound interest. "' i). What are the amount and compound interest of £^2 4s. ^d. for 3 years, at £2 \0s. per cent, halt- l ^l^l t''' ^^^ ''• ^■^'^' '^ *^« ^^riouxii- and JC5 -^s. lUif/., the compound interest. r.o^"i ^^'^^^^ ^^^ *^^" amount and compound interest of ^971 0,; 2\d. for 13 y.^ars, at 4 per cent, per annum } Ans in 616 15.S-. 115^^. is the amount; and £645 Ids Jid.j the compound interest. 24. Given the amount, time, aud rate— to find the pnncipal ; that is, to find the present icorth of any sum to be cire hereafter— a certain rate of interest being allowed for the money now paid. lluLE.— Find the product of as many factors as there are tunes of payment— each _ of the factors bein^ the amount of £\ for a single payment ; and divide this proau<"t mto the given amount. Example.— What sum would produce £834 in 5 years, at per cent, compound iateresf? TJio amount of XI for 1 year at the given rate is £1-05 ; }o-Xl'^^^,^'^''^X^"0^' ^^^"«^ (according to the table) is i:831-M-27G28=i:G53 9s. 2,^./., the required principal. 25. Reason- of thk Rui.k.—Wc have seen [21] that the nmoxmt of nny sum is equal to the amount of &\ (for the sanu tm.e, and at the same rate) multiplied by tlie principal ; that is, the !;L;uroV,£ L ''' ^'^^■^^ pHnoipal==thc given ^incipalX r,nl^JI.%J'™v ^'w,'^' ^''''? '^"''^ quani.tics by the same number [bee. V. b], the quotients will be equal. Tl.croforc— _ Ihe amount of :'.o given principal -f- the amount of £l=thc given prmcipalxthe amount of £l-Mhe amount of &\. Tliat r' • 1 ] T^T^ ""^ ^^"^ ■?'^'"'' princiijal (the given amount) divided by the amount of £1, is equal to the principal, or quantity required— which is the rule. i ^^ »^ EXERCISES. 7. What ready money ought to be paid for a debt of X629 176-. \\\d., to be duo 3 years hence, alluwin--' S i)er cent, compound interest } Ans. j£^500. ° INTRRERT. 257 8<f., H. mat principal, put to interest for G years, would 0. What sum u'ould produce ^£742 Ids UUl in 14 years, at 6 per cent, per annum ? Ans. ^32s' 12.. 7d. 10. ^^ liat IS .€495 19.. ll|r/., to be due in IS years, il71 IT'sfd ^^'^^"^'*^^"^'^^' ^^'^^th at present. ^^«.: tlio^timo^*'" ^'"^ P''"'cip<'^^ rate, and amount— to find ^ Rule I.—Divide the amount by the principal: and mto the quotient divide the amount of £1 for one my^ ment (at the given rate) as often as pcssible-the number ol times the amount of £1 U, been used as a divisor, will be the required number of payments. 0\i''il'T'~^'' jvhat time Avill £92 amount to £100 13.. V^d., at 3 per cent, half-yearly '? XlOG 13.. O'v/.^.:e92=ll.o927. The amount of £1 for one payn.ont i. ,£103. Vnit 115927 -- 103 - 11255 • 1 • 12.),) -^ 103 == l-n'J272 • I0'i'^7'> • 10'^ — i n^^o A 10009^103^1.03; lS3il^j:;i-\(;eIale^.i t^ as a divisor tiinco; therefore the time is 5 pavmenta. or S^^vTo-^r'^""'^ *''"'" ^''^^ ^'" ^ ron)ainder after dWid- Dip, \>y i Uo, tec. as often as possible In explaining the method of tinding the powers and roots moni7n?n'^"^''f •^■'- ''i "^'^'^ hereafter, notice a shorter method of aseertan.ing ],cnv ofton the amount of one pound can Ije used as a divisor. '■ 27. Rule H.—Divide the given principal by the given amount, and ascertain by the interest table in how iiumy pnyments £1 would be equal to a quantity nearest to the quotient— considered as pounds : this will be tho rcquured time. KxAMPi.K.—In what time will £50 become £100, at G per cent, per atmum compound interest ? £100-1-50=2. ^i^«o^^o''M''^ 1^'? ^f^'^'' *'^''* ^" 1^ years £1 will become ro moo '1'-1' ■' ^^'' ' '^'"^ ^" ^- y^^'-' ''"It it ^^'ill l^ecome ii '? .; ^; ""^\.'''^ ^^y^^^^ than 2. Tho answer nearest to the truth, thereluro. is 12 years. li^Si ^8 INTKUr.ST, 28. Rkason ok lliTLK I. — TIio given amount is [20] cquul to the givon principal, nuiltiplioil by a proiluct wlucii containa as many factors as there arc tijiies of payment— each factor being the anioiint of i^l.fur one payment. Hence it is evi- dent, that if we divide the given amount by tlie given prin- cipal, we must have the product of these factors ; and tiiat, if we divide this product, and tlio succesHivo quotients by one of the factors, wc shall ascertain their number. l^EAsoN OF llui.E 11. — We can find the required number of factors (eacli tlie amount uf £1), by ascertaining how often the amount of .£1 may bo considered us a factor, withuut forming a product tmich greater or less than the quotient obtained when we divide the given amount by the given principal. Instead, however, of calculating for ourselves, we may have recourse to tables constructed by those who have already made the necessary multiplications — which saves much trouble. 29. When the quotient [27 J is greater than any amount of £\^ at the given rate, in tlie table, divide it by the greatest found in the table ; and, if necessary, divide the resulting quotient in the same way. Continue the process until the quotient obtained is not greater than the largest amount in tlie table. Ascertain wliat fimiber of jxnjments corresponds to the last quotient, and add to it so many times the largest nniuler of pay- vients in the table, as the largest amount in the table has been used for a divisor ExAMPLK. — When would £22 become X535 12s. O^d., at 3 per cent, per annum '? £535 12s. OJri.-^ 22=24-34500, which is greater than any amount of £1, at the ^iven rate, contained in the taljle. 24-34560-f-4-383l) (the greatest amount of £1, at 3 per cent., found in the ta1)lc)=5'55339 ; but this latter, also, is greater tluxn any amount of £1 at the giv<-n rate in tlie tables. 5-55339-i-4'383'J=l'2(iG77, which is found to bo the amount of £1, at 3 per cent, per payment, in 8 payments. We have divided by the highest amount for £1 in the tables, or that corresponding to iifty payments, twice. Therefore, the required time, is 50-j-50-f-8 payments, or 108 years. EXERCISES. 11. When would £14 6^. 8^. amount to i218 2s. 8^d. at 4 per cent, per annum, compound interest ? Ans. In 6 years. INTEUKST. 259 12. Wlion would jer)4 25. 8^/. amount to £76 35. 5d.^ Hi 5 per cent, per annum, compound interest .'* Ans. Tn 7 years. 13. In wliat tinu! would £793 ().?. 2]f/. become J21034 135. IOJyZ., at 3 per cent, half-yearly, compound interest ? Ans. hi 4^ years. 14. '\Vhcn would £100 become £1639 7.?. 9J., at 6 per cjLt. halt-yearly, compound interest .'' Ans. In 24 years. QUESTIONS. 1. What is interest .? [IJ. 2. AVliat is the diffLjrenco between simple and com- pound interest ^ [1]. 3. AVhat are the principal, rate, and amount ? [1]. ■ 4, How is the simple interest of any sum, for 1 year, found.? [2 &c.]. 5. How is the simjilo interest of any sura, for several years, found } [5]. 6. How is the interest found, when the rate consists of more than one denomination ? [4]. 7. How is the simple interest of any sum, for years, months, &c., found ? [6]. 8. How is the simple interest of any sum, for any time, at 5 or 6, &c. per cent, found .? [7]. 9. How is tlie simple interest found, when the rate, number of years, or both arc expressed by a mixed number ? [9J. 10. How is the simple interest for days, at 5 per cent., found .? [10]. 1 1 . How is the simple interest for days, at any other rat«, found ? [H]. 12. How is the simple interest of any sum, for months at f) per cent., found t [12]. 13. How is the interest of money, left after one or more payments, found ? [13]. 14. How is the principal found, when the amount, rate, and time are given .'' [14]. 1"), How is the" amount found, when the time, ratn, and principal arc given ? [15]. ■hM 2G0 DISCOUNT. 10. ITaw k tliG timo r.uiad, when the amount, prin cipal, II ud rate are given ? [10 J. 17. How is the rate found, when tlio amount, priuci pal, and timo arc given ? [18]. 18. How are the amount, and compound interest found^ wlion the principal, rate, and time are given ? [iDj. I'J. llow is the present worth of any sum, at com- poiuid interest for any time, at any rate, found > [24|. 20. How is the time found, wlirm t)ie principal, rate of compound interest, and amount are given .? [26j. DISCOUNT. 30. Discount is money allowed for a sum paid before it is due, and should be such as would be produced hy what iii paid, were it put to interest from the time the payment is, until the time it ou,o-/ii to he made. The presoit loorth of any sum, is that which would, at the rate allowed as discount, produce it if put to interest until the sum becomes due. 'M. A bill is not payable until thi-ee days nfter the time mentioned in it ; those are called days of grace. TIuis, if the time expires on the 11th of the month, the bill will no^. be payable until the 14th— except the latter falls on a Sunday, in which case it boconies payable on the preceding Saturday. A bill at 91 days will not be duo until the 04th day after date. 32. WHicn goods are purchased, ascertain discount is oft^Mi allowed for prompt (immediate) payment. The discount generally take;, is larger than is sup- posed. Thus, lot what is allowed for paying money one year before it is duo be 5 per cent. ; in°ordinary circumstances ^£95 would bo tho payment for .£100. But £\)b would not in one year, at .5 per cent., produce more than i299 15.v., which is less than £100 ; the eri-or, however, is inconsiderable when the time or sum is small Hence to find the discount and present worth at any rate, we may ge.nerallu use the following — DItCOUNT. 901 2?.. Hulk. — Find the interest for the sura to be paid, at tlu; dificouut uUowod; consider this ns discount, and (hduct it from wha( is due ; the romaindt • will be tlus required present woiLh.- Example. — £<V"' will ^ ; duo in 3 months ; what should b* allowed on inui, lo payment, the discount being at the rato of G per cent, per annum 1 The intorest on £(i2 for 1 year at G per cent, per annum is cC3 1 ' 4'l(l. ; and for 3 months it is IHs. l^d. Therefore jEi02 miuua iSs. 7|ti.=J(iGl Is. 4^(i., u the required present worth. .34. To find the present wortli acrMrakly — ]Ii;le. — Say, as .£100 plus its interest for the given tiuio, is to iilOO, ^ ) is the given pum to the required present worth. TCxAMPLE. — What wnuld, at present, pay a debt nf XI 12 to he due in (J months, b per cent, per annum disooont being allowed ? jC £ £ s. £ £ i()f) V 142 ^ •'• ^• 102-5 (100 f-2 10) : 100 : : 142 : — xi^4~=^^^ ^^ ^ This is merely a question in a rule already given [14]. .1 I i EXERCISES. 1. What is the present worth of ^2850 15i-., payable in one year, at G per cent, discount > Ans. £802 lis. lO^d 2. What is the present worth of £240 10.?., payable in one year, at 4 per cent, discount ? Ans. £231 5^. 3. What is the present worth of £550 10s., payable in 5 years and 9 months, at 6 per cent, per an. discount > Ans. £409 55. loyi. 4. A debt of £1090 will be due in 1 year and 5 months, what is its present worth, allowing 6 per cent, per an. discount ? Ans. £1004 12s. 2d. 5. What sum will discharge a debt of £250 175. 6c?., to bo due in 8 months, allowing 6 per cent, per an. discount .=" A71S. £241 45. 6 J- J. 6. Wiiat sum will discharge a debt of £840, to be duo in 6 montlis, allowing 6 per cent, per an. discount ? A71S. £815 IO5. HUl IMAGE EVALUATION TEST TARGET (MT-2) 1.0 .1 9^ IM 111112 2 If 1^4 lit - 1^ lilM il.25 1.4 1.8 1.6 <^ /a '-e. ^/^ 'V/ ■'^A <Pl ^7 ^"-4 # !m. Phntnorpnlrlr Sciences Corporation 23 WEST MAIN STREET WEBSTER, N.Y. 14580 (716) 872-4503 # :1>^ \ :\ "Q \ « <f ^ # ^'""^«?:^ ^^^' .: W ■ o V <^ ^ #/ 4f^ «. i^ ^ ^ :\ \ 6^ 262 15ISC0UNT. 7 What ready money now will pay a debt of £200, to bo due 127 days hence, discounting at 6 per oent per an.? Am. £ldo iSs. 2-}d. f « ut. 8.^ \Vhat ready money now will pay for ^1000, to be 9 A bill of £150 105. will become due in 70 days what ready money will now pay it, allowing 5 per cent per an. discount ? Ans. dei49 Is. bd 10. A bill of £140 10,. will be due in 76 days, what ready money will now pay it, allowing 5 per cent, per an. discount > Ans. £139 1,. O^'Z. ^ 11. A bill of £300 will be due in 91 days, what wiU ^oTu?^^ ^\ al owing 5 per cent, per an. discount .? Ans. jb29D <ys. l^d. 12. A bill or £39 5^. will become due on the first ot beptember, what ready money wUl pay it on the ^^£38 iS 1-5"^^' ^^^''''''° ^ ^'' "'''*• P"" ^"- • 13. A bill of £218 3.9. SicZ. is drawn of the 14th August at 4 months, and discounted on the 3rd of Oct • what IS then its worth, allowing 4 per cent, per an! discount .? Ans. £216 Ss. \id. 14 A bill of £486 185. 8^. is drawn of the 25th March at 10 months, and discounted on the 19th June what then is its worth, allowing 5 per cent, per an' discount.? Ans. £412 9s. U^d. e ■ 15. What is the present worth of £700, to be due in 9 months, discount being 5 per cent, per an. > Ans £674 135. 11^^. ^ p u. . Jins. 16. What is the present worth of £315 12, 41^/ payable in 4 years, at 6 per cent, per an. discount"? Ans. £254 lO.e. 7]-d. 17 What is the present worth and discount of £550 105. for 9 months, at 5 per cent, per an. } Ans. £530 125. Q\d. is the present worth; and £19 175. lli^; s the discount. * * 18. Bought goods to the value of £35 135. 8^. to be Daul m 294 days; what ready money are they now ivorth, 6 per cent, per an. discount being allowed ? Ans. £31 05. 9^^/. ^ COMMISSION. 263 19. If a legacy of £600 is left to me on the 3rd of May, to be paid on Christmas day following, what must I receive as present payment, allowing 5 per cent, per an. discount.^ Ans. i:i581 4s. 2}d. 20. What is the discount of £756, the one half pay- able in 6, and the remainder in 1.2 months, 7 per cent, per an. being allowed ? Ans. £37 I4s. 2\d. 21. A merchant owes £110, payable in 20 months, and £224, payable in 24 months ; the first he pays in 5 mouths, and the second in one month after that. What did 1)0 pay, allowing S per cent, per an. } Ans. £300. QUESTIONS FOR THE PUPIL. 1. What is discount .? [30]. 2. Wliat is the present icorth of any sum } [30]. 3. \s'\\iii QXQ. days of grace] [.?1]. 4. How is discount ordinarily calculated } [33] 5. How is it accMrately calculated > [34] . COMMISSION, &c 3.5. Commission is an allowance per cent, made to a person called an agent., who is employed to sell goods. Insurance is so mucli per cent, paid to a person who undertakes that if certain goods arc injured or destroyed, he will give a stated sum of money to the owner. Brokerage is a small allowance, made to a kind of agent called a broker, for assisting in the disposal of goods, negotiating bills, &c. 36. To compute commission, &c. — Rule. — Say, as £100 is to the rate of commission, so Is the given sum to the corresponding commission. ffc- Example. — What will be the commission on goods worth £437 56-. 2</., at 4 per cent. ] £100 : £4 : : £437 5s. 2d. : l^l^iJIii-?^ = £17 9^. 100 9i'<Z., the required couimissiion. V7.V 37. To find what insurance must be paid so that, if the goods are lost, both their value and the insurao'io paid mr.y be recovered — 264 COMMISSIOI* Rule.— Say, as £100 minus the rate per cent, is to eClOO, so is the value of the goods insured, to the required insurance. Example.— What sum must I insure that if goods worth i.4UU are lost, I may receive both their value and the in- surance paid, the hitter being at the rate of 5 per cent '* £95 : £100 :: £400 : ^122^0=^421 1. 0^^ If £100 were insured, only £95 would be actually received, since £5 was paid for the £100. In the example, £421 Is Ohd are received; but deducting £21 Is. OU, the insurance, £400 remains, EXERCISES. 1. What premium must be paid for insuring goods to the amount of £900 15s., at 2^ per cent, f A7is £2,2 105. 4ir/. ' - f 2. What premium must be paid for insuring goods to the amount of £7000, at 5 per cent. ? Ans. £350 3. What is the brokerage on £976 175. 6d., at 55. per cent. > Ans. £2 Ss.'lQid. 4. What is the premium of insurance on goods worth £2000,^ at H per cent. ? Ans. £150. 5. ^Vlmt is the commission on £767 145. 7d , at 2i pcrcent. .? A7is. £19 3s. lO^d. ^ y,^',^^'''^ ^'^'^^ ^^ *^^e commission on goods worth i'J71 145. 7rf., at 5.5. per cent. ? Ans. £2 8s. 7-^-d 7. What is the brokerage on £3000, at 25. 6^. per cent. ? Ans. £3 15s. ^ S How much is to be insured at 5 per cent, on goods worth £900, so that, in case of loss, not only the value ot the goods, but the premium of insurance also, may bo repaid ? ^ Ans. £947 75. 4/-^. ' -^ 9 Shipped off for Trinidad goods worth £2000, how much must be insured on them at 10 per cent., that in case of loss the premium of insurance, as well as their value, may be recovered ? Ans. £2222 45. dhd. QUESTIONS FOR THE PUPIL. 1 What is commission ? [35], 2. What is insurance ? [35]. 3. What ia brokerage ? [35] PURCHASE OF STOCK. 265 0?rf. at24- 4. IIow are commission, insurance, &c., calculated? [36]. 5. How is msnrancG calculated, so that both the in- surance and value of the goods may be received, if tho latter are lost ? [37] . PURCHASE OF STOCK. .>o. Stock is money borrowed by Government from individuals, or contributed by merchants, &c.,^ for the purpose of trade, and bearing interest at a fixed, or variable rate. It is transferable either entirely, or in part, according to the pleasure of the owner. If the price per cent, is more th;in £100, tlie stock in question is said to be ahave^ if less than i^lOO, helow " par." Sometimes the shra-es of trading companies are only gradually paid up ; and in many cases the whole price of the sliare is not demanded at all — they may be ^£50, £100, &c., shares, while only £5, £10, &c., u.ay have been paid on each. One person may have many shares When the intesest per cent, on i\\Q money paid is con- .sidera1)lo, stock often sells for more than what it origi- nally cost; on the other hanu, when money becomes more valuable, or the trade for which the stock was contributed is not prosperous, it sells for less. 39. To find the value of any amount of stock, at any rate per cent. — EuLE. — Multiply the amount by the value per cent., and divide the product by 100. ExAMPi.K.— When £G'J \ will purchase £100 of stock, what will purchase £G42 ? £G42x69j 100 -=£443 15s. lid. It is evident that £100 of stock is to any other amount of it, as the price of tho former is to that of tho hxttor. Tims £100 : £612 :: £69 .\ : ^il^^.^ 100 EXERCISES. 1. What must be given for £750 16i\ in the 3 per cent, annuities, when £64 j- will purchase £100 .? Ans. £481 95. O^^V^ ■I iii*( 2Rf) EQUATION OP PAYMENTS. 2. What must be given for ^£1756 Is. 6d. India stock, when £]U6l will purchase dt^lOO ? Ans. £3446 17s. i<^d 3. What is the purchase of ^29757 bank stock,' ai J212oA per cent. ? Ans. jei2257 4^. 7^d. QUESTIONS. 1. What is stock .? [38]. 2. When is it above, and when below " par" ? [38], 3. How is the value of any amount of stock, at an^ rate per cent., found ? [39]. EQUATION OF PAYMENTS. 40. This is a process by which we discover a time, when several debts to be due at dij'ereni periods maybe paid, (il once, without loss either to debtor or creditor lluu.:.--Multiply each payment by the time which should elapse before it would beco-ie due ; then add the products together, and divide their sum by the' sum 01 the debts. Example 1. -A person owes another £20, payable in 6 months; i 50 payable in 8 months; and X90 payable in U months. At what time may all be paid together, without OSS or gam to either party '? * o ) il jl 20 X 0= 120 /30x 8= 400 _90x 12=1 080 IGU 1GO)TOOO(10 the required number of mor'^g. 160 . ExAMPLK 2.— A debt of £450 is to be paid thus : £100 immediately, £300 in four, and the rest in six months \V lien Bhould it be paid altogether ? £ 100 300 _50 450 £ X 0= X 4=1200 X 6=^03^ 450)1500(31 months 1350 "iso 450 EQUATION OP PAYMKNTS. •267 41. Wo liavG (according to a i^rinciplo fonnorly used [13]) reduced each debt to a sum which would bring the same interest, in one month. For G times i^20, to be due in 1 month, should evidently produce the fsame as £20, to be due in G months — and so of the other debts. And the interest of j£lGOO for the smaller time, will just be equal to the interest of the smaller sum for the larger time. EXERCISES. 1. A owes B jeeOO, of which £200 is payable in 3 months, £150 in 4 months, and the rest in 6 months ; but it is agreed that the whole sum shall be paid at once. When should the payment be made ? Ans. In 41 months. 2. A debt is to be discharged in the following man- ner : I at prcijent, and ^ every three months after antil all is paid. What is the equated time ? Ans. A\ months. 3. A debt of £120 will be due as follows : £50 in 2 months, £40 in 5, and the rest in 7 months. "When may the whole be paid together 't Ans. In 4^ months. 4. A owes B £110, of which £50 is to be paid at the end of 2 years, £40 at the end of 3^, and £20 at the end of 4-1^ years. When should B receive all at once .'' Ans. In 3 years. 5. A debt is to be discharged by paying ^ in 3 months, i- in 5 months, and the rest in 6 mouths. What is the equated time for the whole .'' Ans. 4| months. QUESTIONS. 1. "What is meant by the equation of payments } 2. What is the rule for discovering when money, to be due at different times, may be paid at once } [40] . ! ■ ' ii i. 208 SECTION VIII. EXCHANGE, &c. 1. Exchange enables us to find what amoimt of the inoncy of one country is equal to a given amount of the money of another. Money is of two kinds, real— or coin, and imaginary— or money of exchange, for which there is no coin ; as, lor example " one pou7id sterling." The par of exchange is that amount of the money of one country aduallt/ equal to a given sum of tho money of another ; taking into account the value of the metals they contain. Tho mirse of cxchan<rc if^ that sum which, in point of fact, would be allowed for it. 2. When the course of exchange with any plac6 is a^ove ;' par," the balance of trade is against that place. Thus if Hamburgh receives merchandise from London to the amount of ^£100,000, and ships off, in return, goods to the amount of but c£50,000, it can pay only half what It owes by bills of exchange, and for the remainder must obtain bills of exchange from some place else, giving for them a premium— which is so much lo?.t. IJut the exchange cannot be much above par, since, if the pre- mium to bo paid for bills of exchange is high, tho merchant will export goods .it loss profit ; or Tie will pay the expense of transmitting aixl iusuriu*' coin, or bullion. ° 3. The nominal value of commodities in these countries was from four^ to fourteen times less formerly than at present ; that is, the same aiJ^ount of money would then buy much more than now. We may estimate the value of money, at any particular period, from the amount of corn It would purchase at that time. The value of money fluctuates from the uature of the crops, the statu of trade, &o. KXCHANGE. 209 111 ; a.s, la cxcliango, a variublo is given for a fixed sum ; ihiin LonJf)!! receives difioreut values for £1 from diliereut countries. Agio is the dilTerence wbieli there is in some places between the cwrreiU or msk money, and the uxkange or hank money — which is finer. The following tables of foreign coins arc to be mad'.' familiar to the pupil. FOREIGN MONEY. MONEY OF AMSTERDAM. Flemish Money. • . make 1 groto or penny. • . • 1 stiver. • 1 florin or guilder Penningii 16 or 320 800 1920 giote« 40 or 100 240 stivers 20 50 or 120 or guilders 2i 6 1 rixdollar. 1 pound. rfenningg 6 72 or M40 g rotes 12 MONEY OF HAMBURGH. Flemish Money. I • . make 1 grote or penny lings 1 skilliug. 1 pound. Ffenmnga Tenco 12 or 2 192 884 676 I skilli 240 orf 20 Ilamhv,rgh Money. make 1 scliilling, equal to 1 stiver 1 mark. 32 or 64 96 schillings 16 32 or 48 or marks 2 3 1 dollar of exchange. 1 rixdollar. We find that 6 scliillings=l skilling Hamburgh money is distinguished by the word " Harabro." " Lub," from Lubec, where it was coined, was formerly used for tliis purpose ; thus, '• one mark Lub." Wo exchange with Holland and Flanders by the pound Bt^cling. N J-iU rM i>70 KXCIIAWaE. KRKJSCn MONKY. Dernioi!* 12 Accouula wcro Ibnuerly kept in livrus, &c. 210 or 720 Centimes 10 sous 20 make 1 sou. 1 livre. livres 60 or I 8 1 ecu or crown AcoountH are now kept in francs and centimes. • . make 1 dccime. dccimei 100 or I 10 . 81 livrea=80 franca. 1 franc. llocs 400 1000 or 4800 PORTUGUESE MONEY. Accounts are kept in milrees and recs. «... make 1 crusado. crnsadoi 2i . 12 ... . 1 milree. 1 moidore. SPANISH MONEY. Spanish money is'of two kinds, plate and vellon ; the latter being to the former as 32 is to 17. Plate ia used in exchange with us. Accounts are kept in piastres, and maravedi. Maravedies 84 make 1 real. 272 or 1088 375 reals 8 piastres 32 or I 4 . 1 piastre or piece of eight 1 pistole of exchange. . 1 ducat. AMERICAN MONEY. In some parts of the United States accounts are kept in dollars, dimes, and cents. Cents 10 . . , , , make 1 dima idimea 10 . . . . . 1 dollar. In other parts accountg are kept in pounds, shillings, and pence. Those are called currency, but they ar« of much less yalue than with ua, paper money being usad. Pf<«nninjj!i 12 KXCHANtiS DANISH MONET. i?71 make 1 skill! u;;. 102 or nkillin 10 K» marki 1152 OC or (} . amburgU marks. VKNETIAN MONEY. Dnnari (the plurnl ordqnaro) 12 .... innke 1 soldo, soldi 210 or JiO . liro golili 1188 121 or "6 4 l'J20 100 8 . . 1 mark. 1 rixdollar 1 lira. 1 ducftt current. 1 du(^t ellcctivo AUSTRIAN MONEY. rfcmiiiigs 4 210 or 800 Grains 10 oroutzers 00 __. fioring 90 or I U NEAPOLITAN MONEY. cailius 100 or I 10 mako 1 croutzsr 1 florin. 1 rixdollar. make 1 carlin. 1 ducat rtt,A9 MONEY OF GENOA. Lire soldi 4 nnd 12 make 1 scudo di cambio, or crown of exchange. 10 nnd 14 1 scudo d'oro, or gol I crown. Dcnari di pe/.za 12 OF GENOA AND LEGHORN. mako 1 soldo di pozza. I soldi di pez/a 20 f)cnarj di lira 12 240 or 1380 soldi di lira 20 . 110 or 1 5| SWEDISH MONEY. 1 pezza of 8 reals, make 1 soldo di lira. Fonnings, or oers 12 Iskillingi 48 1 lira. 1 pezza of 8 reals make 1 skilling. 1 rixdoUai I f ■ ft! ill ii i I t73 CXOIANOE. RUSSIAN MONEY. 160 mnko 1 ruble. EAST niuko 1 rupco. DIAN W0NE1 Towriei Kiinoci 100,000 . . . iiao. 10,000,000 . . 1 croro. The cowrie is a small ehoU found at the MaMivuH, and near Anj;ola : iu Africa about 5000 of them pass for a pound. The rupcq lias different values : at (Juloutta it is 1;». 11 j,/. tbo Sicca rupee is 2s. OU. ; and the current rupee 2.*.— if wo divide any number of tlicse by 10, we change them to pounds of our money; the Boinbny rupee is 2s. {5^/., &c. A sum of Indian money is expressed as follows; 5-88220, which means 5 la(!3 aiid«8220 rupees. d. To rcduco bank to current money — Ki'i.K.— Say, as J2100 is to JL'IOO + the agio, so is the given amount of bank to tlio recjuired amount of current money. KxAMi'LK. — How many c;uildor.«<, current munoy, arc equal to 403 ouildors, 3 stivers, and 13!jt ponnings banco, a^io being 4^' < .0 a 1<J'^ : lO^ : : 403 g. 3 st. 13«4 p. : 1 ' 7 7 20 ^"^ TOO 05 733 1)203 stivers. 10 45500 14S221 pcmiinga. Multiplying by 05, and adding 04 to tho will give 0034429 I'l'^^li'^t, ]Multi])lying by 733 and dividing by 45500)T0020lio4r)7 will give 155200 penniugs. 10)155209 20 )9700 9 And 485 g. O"^ d^tH p. is the amount sought. 5. We multiply the first and second terms by 7, and add tha numerator of the fraction to one of tlie products. This is tlie same thing as reducing these terms to fractions liaving 7 for their denominator, and then multiplying them by 7 [Sec. V. 29] For the same reason, and in tho same way, we multiply the first and third terms b/ 65, to banish tho fraction, without aeitroying the proportion. eXCHANOE. 273 TIic remainder of tho process i« nccording to Iho rulo of lnoporUofi [Soo. V. 1)1]. VVc roduco tho nnawcr to pcnaingH, BtiverH, mill jjniMoia. • EXKHCIHF.S. 1. R<m1iico n7-l ^•uildofH, 12 Htivors, banlf monoy, to cm rent money, agio being 4i per cent. ? Am. 31)2 g., 5 St., :},V, p. 2. llecluce 4378 guililers, «< stivers, bank money, to current money, agio being 4* per cent. ? Avs. 4577 «., 17 St., r^Vs p. 3. Ueduee S73 guilders. 1 1 stivers, bank money, to current money, agio being 4 J per cent. ? A7is. UIG g., 2st., HJap. 4. llediicc 1012 guilders, bnnk monoy, to current money, agio being 4|i per cent. ? Am. 1722 g., 14Ht., lOA p. 6. To reduee current to Itank money — _ liiu.i:. — Say, as JUlOO-f-thc agio is to JCIOO, so Is tlio given amount of current to the required amount of baidc money. ExAMi'i.K. — How much bank money is thcro in 485 guil- ders and ^J'ii'iol pennings. agio being 4^' i 104? 7 733 4550c 100 7 700 g. St. p. 20 33351500 yioo 10 15520!) Multiplying by 45500 tho denominator, 7002009500 and adding 25957 tho numerator, we get 7002035457 700 33351500)4943424819900 Qu(7tIenl~lT822 1 fl 10) 1 48221;; I 20)9203 403 3 13^^ is tho amount soug*i! i I 274 EXCHANttE. EXERCISES. 5- Reduce 58734 gi^lders, 9 stivers, 11 penningR, current money, to Lank money, agio being 4^ per cent. ? Ans. 560P6 g., 10 St., llJfi p. 6. lieduce 4326 guilders, 15 pcnnings, current money, to bank money, agio being 4-f per cent. ? Ans. 4125 g., 13st.,2i||p. 7. Eeduce 1186 guilders, 4 stivers, 8 pennings, cur- rent, to bank money, agio being 4f per cent. } Ans 1136 g., 10st.,0iff p. 8. Keduce 8560 guilders, 8 stivers, 10 pcnnings, current, to bank money, agio being 4i per cent. . Ans. 8183 g., 19 st., 5fi3. p. 7. To reduce foreign money to Ikitish, &c. — BuLE. — Put the amount of British money considered in the rate of exchange as third term of the proportion, i^^' value in foreign money as first, and the foreign money to be reduced as second term. Example 1. — Flemish Money. — How much British money is equal to 1054 guilders, 7 3tiy~-s, the excliance bcine; 33*-. 4d. Hemish to £1 British 1 S3.S. 4.,;. : r054 g. 7 st. : : £1 : ? 12 20 4U0 pence. 21087 stivers. 2 400)42174 Flemish pence. _£10r,435 = £105 8s. 8iJ. £1, the amount of British money considei'cd in the rate, is put in the third term , 335. 4d.. its value in foreign money, in the first; and 1054 g. 7 gt . the money to be reduced, in the second. 9. How many pounds sterling in 1680 guilders, at 335. 3d. Flemish per pound sterling ,? A71S. JE168 8s. 10. Reduce 6048 guilders, to Rritisli money, at 33.?. \\d. "Flemish per poun'i British .? Ans. i.'594 7a-. -It T-» 1 XI. jL'teuuco money, at 34^. £198 85. 61 f^^, W.7 'Jit, 04S guilders, L. sUveis, to British Flemish per pound sterling > Am M'r EXCHANGE. 375 lit. npw many pounds sterling in 1000 guilders, 10 stivers, exchange being at 335. 4d. per pbuud sterling ? Jbis. iiilOO Is. * Example 2. — Hamburgh Money. — How much British money is equivalent to 476 marks, 9 skillings, the exchange being 33s. iid. Flemish per pound British '? s, d. m. 8. 33 6 : 476 9| : : £1 : I 12 32 2 ling > 16. Reduce 402 grotes. 15232-f 19'=15251i grotes. 402 )152511 £37-9386=£37 I85. 9d. Multiplying the schillings by 2, and the marks by 32, reduaes both to pence. 13. How much British money is equivalent,to 3083 marks, 12| schillings Ilanibro', at 325. 4d. Flemish per pound sterling .? Ans. £254 65. 8^^. 14. How much English money is equal to 5127 marks, 5 Schillings, Hambro' exchange, at 36s. 2d. Flemish per pound sterling ? uins. i£378 Is. 15. How many pounds sterling in 244S marks, 9|- schillings, Hambro', at 32^. 6d. Flemish per pound ster- A/is. £200 105. 7854 marks, 7 schillings Hambro*, to British money, exchange at 345. lid. Flemish per pound sterling, and agio at 21 per cent. ? Ans. £495 Ids. Old Example 3. — French Money. — Reduce 8654 francs, 42 centimes, to British money, the exchange being 23f., 50c., per £1 British. f. c. f. 0. 8654-42 23 50 : 8054 42 : : 1 : ■-23:50=-=^368 5s. 5^. 42 centimes are 042 of a franc, since 100 centimes make 1 franc. 17. Reduce 17969 francs, 85 centimes, to British money, at 23 franc \, 49 centimes per pound sterling > Ans. £765. 18. Reduce 7672 francs, 50 centimes, to British money, at 23 francs, 25 centimes per pound sterling ? Ans. £330. 276 EXCHANGE. 10. EetlucG 1,5647 francs, 36 centimes, to British money, at 23 francs, 15 centimes per pound sWlinf/ ? A71.S. £675 ISs. 2ld. • ^ ^ o 20. lieduce 450 francs, 58^^ centimes, to British money, at 25 francs, 5 centimes per pound sterling > Ans. £nQ Us. ^ ' Example 4. ■ Pmiuguese Money. —RovT much British money is equa to 540 milrees, 420 rees, exchange beino; at OS. m. per milree ? o o m. m. r. s. d. 1 : 540-420 : : 5 G : 540-420x5s. 6c^.=£148 125. 3,^^. ^,V!".^^^^® *^^*^ S>^i*i»h money is the variable quantity, and OS. M. is that amount of it which is considered in the The rees are clianged into the decimal of a milree bv putting them to the right hand side of the decimal point since one reo is the thousandth of a milree. 21. In 850 milrees, 500 rees, how much British money, at 55. 4d. per milree } Ans. £22Q \Qs. 22. Reduce 2060 milrees, 380 roes, to English money at 55. Q^d. per milree } Ans. ^£573 0^. lOi^. 23. In 1785 milrees, 581 rees, how m*any pounds sterling, exchange at 64i per milree.? Ans £479 175. Qd. 24. In 2000 milrees, at 5^. 81^^. per milree, how many pounds sterling.? Ans. £570 165. S^^. Example 5 —5;,_aH?5/i ilfonei/.— Reduce 84 piastres, 6 reals, IJ maravedi, to British money, the exchange beino- 4Ur/ the piastre. ® o • ^"" r. 6 8 m. 19 d. 40 8 34 272 678 reals. 34 23052 maravedi. 49 272)1129548 4152-7, &c.=£17 %:. 02d. EXCHANGE. 277 EXERCISES. 25. ReducG 2448 piastres to British money, exchange at 50^. sterling per piastre ? Ans. i£;510. 26. Keduco 30000 piastres to British money, at 40d. per piastre ? Ans. £5000. 27. Reduce 1025 piastres, 6 reals, 22if ^ maravedi, to British money, at :^9ld. per piastre ? Ans. i2167 15s. 4d. Example 6. — American Money. — Reduce 37G5 dollars to British money, at 4s. per dollar. 4s.=£\ : therefore 5 )37G5 dol. del. s. £ 753 is the required sum. Or 1 : 3765 : : 4 : 753 28. Reduce £292 3.?. 2^d. American, to British money, at 66 per cent. .? Am. £176. 29. Reduce 5611 dollars, 42 cents., to British money, at 4s. d^d. per dolLir ? Ans. £1250 175. 7d. 30. Reduce 2746 dollars, 30 cents., to British money, at 45. S^d. per dollar ? Aqis. £589 Gs. 2^d. From these examplcM the pupil will very easily under- stand how any other hind of foreign, may be changed to British money. 8. To reduce Britisli to foreign money — Rule. — Put that amount of foreign money which is considered in the -rati* of exchange as the thjrd term, its value in British money as the first, and the British money to be reduced as the second term. ExAMPLK 1. — Flemk'i Money. — How many guilders, &o., in jC2oG 149. 2:1. Britisli, the exchange being 34s. 2d. Flemish to £1 British I £ £ s. d. s. d. 1 : 23G U 2 :: 34 2 : ? 20 20 12 20 12 240 4734 12 5G8UM. 410 410 pence. 24Qyr>202100 T2" )97050-4, &c. 20)8087 _ji_ iC404 7 U Flenieh. N 2 278 EXCHANGE. We might take parts for the 34.s. 2d.— 345. 2d.=£l 4- 10s.4-4s.+2c/. £> £ s. d. ^ei = 1 23G 14 2 I0s.= i 118 7 1 47 6 10 5i i 4s.=a ^ 2^^"=rL (aV of 1) 1 19 £404 7 61 Flemish. EXERCISES. 31. In £100 l5., how much Flemish money, exchan<ro at 335. 4d. per pound sterling? Am. 1000 guilders, 10 stivers. ' 32. Reduce £168 85. dji^d. British into Flemish, exchange being 335. 3d. Flemish per pound sterling ? uItw. 1680 guilders. ^ " 33. In £199 ll5. 10^/j^.Britisli, how much Flemish J!!?i!?^' exchange 345. 9^. per pound sterling ? Ans. 2080 guilders, 15 stivers. 34. Reduce £198 85. e^d. British to Flemish inoney, exchange being 345. 5d. Flemish per pound sterling > Ans. 2048 guilders, 15 stivers. in £'9aT''p^-7?""*T^^ JMovj^i/.-How many marks, &c., in £24 65. British, exchange being 335. 2d. per £1 British l £1 20 20 £24 65. 20 486 398 33s. 12 2d. 398 grotes. 20 )193428 2)9671 8 pence. 16 )4835 schillings, 1 penny. 302 marks, 3 schillings, 1 penny. 35. Reduce £254 65. 8d. English to Hamburgh money, at 325. 4d. per pound sterling.? ]Ans. 3083 marks, 12| stivers. 36 Reduce £378 I5. to Hamburg money, at 365 2d. Flemish per pound sterling ? Ans. 5127 marks. 5 schillings. _ ' 37. Keduce £536 to Hamburgh money, at 865. 4d per pound sterling .? Ans. 7303 marks. EXCHANGE. 279 38. Reduce JB495 155, OJ<Z. to Hamburg currency, at 345. lid. per pound sterling ; agio at 21 per cent. ? Ans. 7854 marks 7 schillings. Example S. — French Moncij. — How much French money is equal in value to £83 2s. 2d., exchange being 23 francs 25 centimes per £1 British ^ £ £ s. d. t 1 : 83 2 2 : : 2325 : ^ 20 20 20 1662 12 12 240 19946 23-25 240 )463744-50 19322-7, or 19322f. 70c. is the required sum 39. Reduce £274 55. Oc?. British to francs, &c., ex- change at 23 francs 57 centimes per pound sterling r Alls. 6464 francs 96 centimes. 40. In £765, how many francs, &c., at 23 franca 49 centimes per pound sterling } Ans. 17969 franca 85 centimes. 41. Reduce £330 to francs, &c., at 23 francs 25 cen- times per pound sterlijjg .? Ans. 7672 francs 50 cents. 42. Reduce £734 45. to French money, at 24 franca 1 centime per pound sterling } Ans. 1769 francs 42-J- centimes. Example 4. — Portuguese Money. — How many milrees anrf, rees in £32 6s, British, exchange being 5s. 9(/. British pe milree 1 s. d. £ s. 5 9 : 32 G : : 1000 : ? 12 20 69 646 12 7752 1000 69)7752000 ,«quircd sura 112348 rce8=112 milrees 348 rees, is tno 260 XXCHANGE. 43. Reduce £226 16^. to milrecs, &c., at 5^. 4d. per milrce ? Ans. 850 milrees 500 roes. 44. Reduce £^479 17*. 6d. to milrees, &c., at 6Ud por milrec ? Ans. 1785 milrees 581 rees. ' * 45. Reduce £570 16.9. 8^. to milrees, &c., at 5*. Sid per milree ? Ans. 2000 milrees. 46. Reduce £715 to milrees, &c., at 5*. 8d. permU- ree ? Ans. 2523 milrees 529/^ rees. . ^^£^S^F,.5--'Si'«ww;i Morm/.— Row many piastres, &o., in £02 British, exchange being 50d. per piastre '» d. £ 50 : 62 : : 1 : ? 20 1240 p. r. m. 12 ^97 32if , is the required sum. 50 )14880 ^ 297-6 piastres. 8 48 reals. 34 50)1632 32M maravedis. •*7. How many piastres, &c., shall I receive for £510 sterling, exchange at 50c?. sterling per piastre ? Ans. 2448 piastres. 48. Reduce £5000 to piastres, at 40^. per piastre > Ans. 30000 piastres. 49. Reduce £167 15*. 4d. to piastres, &c., at SO^d. per piastre ? Ans. 1025 piastres, 6 reals, 22^4^ mara- vedis. ' ^ 50,. Reduce £809 95 8d. to piastres, &c., at 40|J. per piastre ? Arts. 4767 piastres, 4 reals, 2yVV maravedis. Example Q.— American Money .—Reduoe £176 British to American currency, at 66 per cent. £ £ £ 100 : 176 :: 166 ; : 166 100)29216 £292 35. 2i(/., is the required sum. EXCHANGE. 281 EXERCISES. 61. Reduce £753 to dollars, at 4s. per dollar > Ans. 3765 dollars. 52. Ileduce ^£532 4s. Sd. British to American money, at 64 per cent. > Ans. £872 175. 3d. 53. Ikduce £1250 17s. 7d. sterling to dollars, at 4$. 5^d. per dollar ? Ans. 5611 dollars 42 cents. 54. Ileduce £589 6s. 2^%d. to dollars, at 4s. S^d. per dollar } Ans. 2746 dollars 30 cents. 65. Reduce £437 British to American money, at 78 per cent. ? Ans. £777 17s. 2^d. 9. To reduce florins, &c., to pounds, &c., Flemish — Rule. — Divide the florins by 6 for pounds, and — adding the remainder (reduced to stivers) to the stivers —divide the sum by 6, for skillings, and double the remainder, for grotes. Example.— How many pounds, skillings, and grotes, in 105 florins 19 stivers '? f. St. 6)165 19 £21 13s. 2d., the required sura. 6 will go into 1G5, 27 times— leaving 3 florins, or 60 stivers, "which, with 19, make 79 stivers ; 6 will go into 79, 13 times- leaving 1 5 twice 1 are 2. 10. Reason of ^he Rule.— There are 6 times as many florins as pounds ; for we find by the table that 240 grotea make £1, and that 40 C^*") grotes make I florin. There are 6 times as many stivers as skillings ; since 96 penniugs^make 1 skilling, and 16 (V) pfennings make one stiver. Also, sinca 2 grotes make one stiver, the remaining stivers are equal to twice iiH many grotes. Multiplying by 20 and 2 would reduce the florma to grotes ; and dividing the grotes by 12 and 20 would reduce thorn to pounds. Thus, using the same example— f. St. 165 19 20 3319 2 12 )6638 - 20)653_ 2 £27 ]3s. 2d., as before, is the result. f ' ?l « 282 EXCHANGE. EXERCISES. 56. Ill 142 florins 17 stivers, how many pounds, &c., Atis. £23 16*. 2d. 57. lu 72 florins 14 stivers, how many pounds, &c., Ans. £\2 2s. 4(1. 58. In 180 florins, how many pounds, &c. } Am. iE30 11. To reduce pounds, &o., to florins, &c. — Rule. — Multiply the stivers by 6 ; add to the producfi half the number of grotes, then for every 20 contained in the sum carry 1, and set down what remains above the twenties as stivers. Multiply the pounds by 6, and, adding to the product what is to be carried from the stivers, consider the sum as florins. Example. — How many florins and stivers in 27 pounds, 13 skillinga, and 2 grotes ? £f s. d. ' 27 13 2 . 6 165fl. 198t., the required sum. 6 times 13 are 78, which, with half the number (f ) of grotes, make "^0 stivers — or 3 florins and 19 stivers (Z twenties, and 19) ; putting down 19 we carry 3. 6 times 27 are 1G2, and the 3 to be carried are 165 florins. This rule is merely the converse of the last. It is evident that multiplying by 20 and 12, and dividing the product by 2 and 20, would give the eamo result. Thus £ s. d. 27 13 2 20 568 . 12 2)6638 20)3319 165fi. IDst, the same result as before. EXERCISES. 59. How many florins and stivers in 30 pounds, 12 skillings, and 1 grote ? Ans. 183 fl., 12 st., 1 g. 60. How many florins, &c., in 129 pounds, 7 skil- linffs ? Ans. 776 fl. 2 st. 61. In 97 pounds, 8 skillings, 2 grotes, how many florins, &c. : Ans. 584 fl. 9 st. ARBITRATIOX OF EXCHANGES. 283 QUESTIONS. 1. What is exchange ? [1]. 2. What is the difference between real and imagin- ary money ? [!]• 3. What are the par and course of exchange ? [IJ. 4. Wliat is agio? [3]. 5. What is the difference between current or cash noney and exchange or bank money ? [3] . 6. How is bank reduced to current money ? [4]. 7. How is current reduced to bank money ? [6] . 8. How is foreign reduced to British money ? [7] . 9. How is British reduced to foreign money ? [8].^ 10. How are florins, &o., reduced to pounds Flemish, 11. How are pounds Flemish, &c., reduced to florins, fee? [11]. ARBITRATION OF EXCHANGES. 12. In the rule of exchange only two places are con- ecfned ; it may sometimes, however, be more beneficial «o the merchant to draw through one or more other places. The mode of estimating the value of the money of any place, not drawn directly, but through one or more other places, is called the arUtration of exchanges^^^ and is either simph or cortipound. It is " simple " when there is only one intermediate place, " compound " when there are 7/wre than one. All questions in this rule may bu solved by one or more proportions. , 13. Simple Arbitration of Exchanges.— Given the course of exchange between each of two places and a thu-d, to find the par of exchange between the former. , i • x Rule.— Make the given sums of money belonging to the third place the first and second terms of the propor- tion ; and put, as third term, the equivalent of what is in the first. The fourth proportional will be the value of what is in the second term, in the kind of mQuey contained in the third term. t 284 ARnrrnATioN of exchanges. ExAMPLK. — If London c-ohanfoa with T'nrm nf in; .^ franc, an.l with Ani.tcnJa.u at 3-1.. njp Vi Z- ul^ wCJ ought to bo tho cour.s« of exoha,.:o,Tctt of p2 t; Amstonhun that a n.erohant n.ay without loss ron^frca L.Midon to Amsterdam through I'aria '? ""cirom Df £lV'.\^n ■• ^"^'- ^'/- ^*K^ equivalent, in Klomi«h money, Fiefuilh mi!;:;'" "' '''■ ^^''^' ^^^ ^' ^ ^'^'^"^) ^^ ^^^- • 240- '^Tl andlO f ^^'h?''f'^^ ?' «f ^ franc, in Flemish nu.ney. that which belongs to the third place; and 34. 8 i/'tho given equivalent of £1. ^ ' ^ -Jii. oa. is tno It is evident that, 17U. (Flemish) bein- the value of in,/ <,i>iitisn;, out lie will not recp vf> ITJ^,; f.n. fi.,*- i v, EXERCISES. is 1^5 o5' '^"^^^"Se between London and Amstordani wh?l;!f'''^"^'' "^^t<^«dto Petorsburgh 5000 ruble. • 7'lt fo/"^:Tu''7^^^^^^ «nd London ih c t oOd per ruble, but between Petcrsburrrh an-l Holland It IS at 90^. Plomish per ruble, and Holl d Which will be the more advantageous method for Lon- ' don be drawn upon-the direct°or the indirect ? Ans Jjondon wdl .o-;iin ]e9 n? 1 en^/ :p u ^ by way of Holland "» '' '^ '' ""*"^ P"^'""'"^ 5000 rnblos— ^1041 T?. 17 p ... , r,,-,-- but ^1875 Fre«i;h=il^32 1 l^lVlufh.'" ^'^""^" ' ARBITRATION OV EXCHANGES. 385 14. Compound Arhitrntion of Exchanges. — To find what should bo the course of exchange between two places, through two or more others,, that it may be on a par with the course of exchange between the same two places, dircdly — IluLE. — Having reduced monies of the same kind to the same denomination, consider each course of exchan<»o as a ratio ; set down the dift'orent ratios in a vertical column, so that the antecedent of the second sliall be of the same kind as the consequent of the first, and the antecedent of the third, of the same kind as the conse- quent of the second — putting down a note of interroga- tion for the unknown term of the imperfect ratio. ThcL divide the product of the consequents by the product of the antecedents, and the quotient will be the value of tho given sum if remitted through the intermediate places. Compare with this its value as remitted by tho direct exchange. 15. ExAMPLK.— £824 Flemish being due to me at Am- sterdam, it is remitted to France at IGrf. Flemish per franc; from Franco to "Venice at 300 francs per GO ducats : from Venice to Hamburgh at lOOtZ. per ducat ; from Hamburgh to Lisbon at 50f/. per 400 rees ; and from Lisbon to England at 5.S-. 8^/. sterling per milrce. Shall I gain or lose, and how much, tho exchange between England and Amsterdam being 34i'. 4t/. per XI sterling ? \^d. : 1 franc. 300 francs.: GO ducats. 1 ducat : 100 pence Flemish. 50 pence Flemish : 400 roes. 1000 rees ; G8 pence BritisJi. '? : £824 Flemish. ^XC0xl00x400x68x824 ,.^ '10X300X1X50X1000 =^'^ '^"^ ''^'^^'° *^^' **^^^"« [Sec. V. 47]) 11^^=£5G0 Gs. A\d. But the exchange between England and Amsterdam fd £824 Flemish is £480 sterling. Since 34s. M. : £824 : : £1 : .^^^'^.^£430. I gain therefore by the circular minus £480=£80 65. Aid. 34.S. 4d. exchange X5G0 G*-. 4|u. 286 AKniTRATION OF RXCnANQES. If commission ia chaxf^d in any of the places, it must bo do(Juct(!d from tho value of tho sum which cuu bo obtained in that place. Tho procoss given for tho compound arbitration of ox- cliiin;5o may bo provo<l to bo correct, by putting down tbo difFerent proportions, and nolving tbcm in Hueeeswion. 'Ibus, if 10.'/. aro equal to 1 franc, what will 300 francs (=00 ducats) bo worth, ff tlio quantity last found is tho valuo of 00 ducats, what will be that of cue dueat (=3l00t/.), &o. '? EXfiKCLSES. 3. If London would remit iDlOOO sterling to Spain, tho direct exchange being 42),(l. per pia.stre of 272 maravedis ; it i.s ankoA whether it will bo more profit- able to remit directly, or to remit first to Holland at 3o5. per pound ; thence to France at Id^d. per franc ; thence to Venice at 300 francs per 60 ducats ; and thence to Spain at 3G0 maravedis per ducat ? Ans. The circular exchange is more advantageous by 103 piastres, 3 reals, lOf^- maravcdLs. 4. A merchant at London has credit for 680 piastres at Leghorn, for which ho can draw directly at oOd. per pia.stre ; but choosing to try tho cii-cular way, they aro by his orders remitted first to Venice at 94 piastres per 100 ducats; thence to Cadiz at 320 maravedis per ducat ; thence to Lisbon at 630 rces per piastre of 272 maravedis; thence to Amsterdam at 5 W. per crusade of 400 rocs ; thence to Paris at IS^d. per franc ; and thence to London at 10^^?. per franc ; how mucb is tho circular -emittanco better than tho direct draft, reckon- ing I per cent, for commission ? Ans. ^£14 12s. l^d 16. To estimate the gain or loss per cent. — lluLE. — Say, as the par of exchange is to the c; urso of exchange, so is iElOO to a fourth proportional. From this subtract £100. Example. — ^The par of exchange is found to be IS^d. Flemi.sh, but tho cour.se of exchange is Idd. per fraiic ; what is the gain per cent. ? £19x100 lo ia. fAOO 'M — =X104 7*-. Ud. Thu.s I X4 Is. IJ If in paid, it i 5. Th but tho < cent. ? 6. Th course ik 6*. lli</ 7. Th course of Ans. £1 1. W] 2. Wl pound ai 3. AVI 4. Wl 5. He any plac 6. Ho 17. T gain or 1 certain ] Given gain or 1 KULE and at tl or loss ExAMr G(i., and i Thetc The tc Thetc I'UOFIT AND LOM. 287 Thus (ho piiu por ccnt.=,C104 7s-. l^^ nilnuH £100=* X4 7v. 11(/. if the merchant remits through I'liria. It' in remitting through Paris oommisHiou must ba paid, it is to be deducted from the gnin. EXERCISES. 5. Tho par of exchange is found to bo \8^d. Flemish, but the course of exchange is 19|t/., whatis the gain per cent. ? Ans. £4 ISs. 2|</. 6. Tho par of exchange is 17 ^d. Flemish, but tho course Is 18|tZ., what is the gain per cent. ? Ans. £4 6s. UU. 7. The par of exchange is 18^^. Flemish, but tho course of exchange is 17|^rf., what is the loss per cent. ? Ans. £1 165. 2d. QUESTIONS. 1. What is meant by arbitration of exchanges .? [12]. 2. What is the difference between simple and com- pound arbitration } [12]. 3. AVhat is the rule for simple arbitration ? [13]. 4. What is tho rule for compound aibitration ? [14]. 5. How arc we to act if commission is charged m any place .? [15]. 6. How is the gain or loss per cent, estimated } [16]. PROFIT AND LOSS. . 17. This rule enables us to discover how much we gain or lose in mercantile transactions, when we sell at certain prices. Given the prime cost and selling price, to find the gain or loss in a certain quantity. KuLE. — Find the price of the goods at prime cost and at the selling price ; the difference will be the gain or loss on a given quantity Example. — What do T gain, if I buy 460 lb of butter at ijd.j and sell it at Id. per lb ? The total prime cost is 460J.x6=2760f?. Tlic total sGlliiig price is lOuCi.X i=o^^Od. The total gain is o220(/. minus 27G0J.=460c/.=jCl 18s. id. r. i ♦ i i 288 PROFIT AND LOSS. % EXERCISES. 1. BougLt 140 ft) of butter, at lOd. per ih, and eold it nt 14d. por ft) ; what was gained ? Ans. £,'1 6s. 8</. 2. Bought 5 cwt., 3 qrs., 14 lb of cheese, at £2 I2s. per cwt., and sold it for d22 185. per cwt. What was the gain upon the ^vholc ? Ans. £1 15s. 3d. 3. Bought 5 cwt., 3 qrs., 14 ft) of bacon, at 345. per What was the cwt. and sold it at 365. 4d. per cwt. gain on the whole .'' Ans. I3s. 8^d. 4. If a chest of tea, containing 144 ft) is bought for 6s. 8ft. per ft), what is the gain, the price received for the whole being £57 10s. } Ans. £9 lOs. 18. 1*0 find the gain or loss per cent. — Rule.: — Say, as the cost is to the selling price, so Is £100 to the required sum. The fourth proportional minus £100 will be the gain per cent. Example 1. — What do I gain per cent, if I buy 1460 lb of beef at 3(Z., and sell it at Z^d. per Bb '^ 3(Z.xl460=4380tf., ia the cost price. And 3i(/.xl4G0=5110rf., is the selling nrice. 5110 X 100 Then 4380 : 5110 : : 100 : — ^^^ — = £116 13s. 4d. Ans. £116 13.S. 4d. minus £100 (=£1^ 135. 4d.) is the gain per cent. REAijON OF THE RuLE. — The price is to the price plus the gain in one case, as the price (£100) is to the price plus the gain (£100-f-the gain on £100) in anotiicr. Or, the price is to the price plus the gain, as any multiple or part of the former (£100 for instamse) is to an equimultiple of the latter (£lOO-f-the gain on £100). Example 2. — A person sells a horse for £40, and loses 9 {)er cent., while he should have made 20 per cent. What ia lis entire loss "? £100 minus the loss, per cent., is 1o £100 as £40 (what the horse cost, minus wliat ho lost by it) is to what it cost. 01 : 100 : : 40 : — — — =£43 19*. liJ., what the horse cost. But the person should have gained 20 per cent., or ^ of the price j therefore his profit tihould have been PROFIT AMI LOSS. 289 £ x. d. 3 19 l.V ia the difference between cost and selling price. 8 15 9^ is what he should have received above cost. 12 14 11} is his total loss. so IS , or \ been EXERCISES. 5. Bought beef at 6(Z. per lb, and sold it at ^d. What what was the gain per cent. } Ans. 331-. 6. Bought tea for' 5s. per lb, and sold it for 3s. What was the loss per cent. > Ans. 40. 7. If a pound of tea is bought for Qs. Qd.^ and sold for Is. 4d.^ what is the gain per*cent. ? Ans. 12ff . 8. If 5 cwt., 3 qrs., 26 lb, are bought for £9 85., and sold for £11 185. 11^., how much is gained per cent. } Ans. 27 ^V^. 9. When wine is bought at 175, Gd. per gallon, and sold for 27.V. 6c/., what is the gain per cent. ? Ans. 57^. 10. Bought a quantity of goods for j£60, and sold them for ^£75 ; what was the gain per cent. .'' Ans. 25. ^11. Bought a tun of wine for £50, ready money, and sold it for £54 IO5., payable in 8 months. How much per cent, per amium is gained by that rate .'' Ans. 13^. 12. Having sold 2 yards of cloth for II5. 6</., I gained at the rate of 15 per cent. What would I have gained if I had sold it for 12?. t Ans. 20 per cent. 13. If when I sell cloth at 75. per yard, 1 gain 10 per cent. ; wh t will I gain per cent, when it is sold for 85. 6i. .? Ans. £33 Us. 5^d. 'Is. : 8.S-. 6(!. •: £110 : £133 lis. 5\d. And £133 II5. 5!/L— £100=£33 il.^ 5 i(Z., is the required gain. 19. Given the cost price and gain, to find the selling price — Rule. — Say, as £100 is to £100 plus the gain per cent,, so, is the cost price to the required selling price. Example. — At what price per yard must I sell 427 yards of cloth which I bought for 19*'. per yard, so that I may gain 8 per cent. 1 lOSxiO.N'. 100 : 108 : : 10s. : — iqq— =JC1 O5. G\d. This result may be proved by the last rule. 290 rnOFIT AND LOSS. EXERCISES. 14. Bought velvet at 4.?. 8f/. per yard ; at what price must I sell it, so as to gaia 12^- per cent. ? Ans. At 55. 3d. 15. Bought muslin at 55. per yard ; how must it be sold, that I may lose 10 per cent. ? Ans. At 4i*. 6d. 16. If a tun of brandy costs £40, how must it be sold, to gain 6i per cent. ? Ans. For j£42 10a\ 17. Bought hops at ii4 165. per cwt. ; at what rate must they be sold, to lose 15 per cent. .? Ans. For £4 Is. l\d. 18. A merchant receives 180 casks of raisins, which stand him in \Qs. each, and trucks them against other merchandize at 28s. per cwt., by which he finds he has gained 25 per cent. ; for what, on an average, did he sell each cask ^ Ans. 80 lb, nearly. 20. Given the gain, or loss per cent., and the selling price, to find the cost price — Rule. — Say, as JGJIOO plus the gain (or as J3100 minus the loss) is to £100, so is the selling to the cost price. FiXAMPLB 1. — If I sell 72 K) of tea at (js. per lb, and gain 9 per cent., what did it cost per Jb ? 109 : 100 : : 6 : — Jq^=5s. M. What produces £109 cost £100 ; therefore what pro- duces Os. must, at the same rate, cost bs. Qd. Example 2. — A merchant buys 97 casks of butter at 30.«. each, and selling the butter at £4 per cwt., makes 20 per cent. ; for how much did he buy it per cwt. ? 30.v.x97=2910s,, is the total price. Then 100 : 120 : : 2910 : -^~?-^= 3492s., the 100 3492s. Belling price. And ~q7)7' \='~£T^ )=43G5, is the number of cwt. ; and -,jy-=50]^* lb, is the uvcrage weight of each cask. lb lb .S. 110 vQ Then 50}lj : 112 : : 30 : li"^'^' : GO*. 8(/. = £3 65. 8(i., the required cost price, per cwt. FELLOWSHIP. 291 EXERCISES. ,19i. Having sold 12 yards of cloth at 20*. per yard, and lost 10 per cent., what was the prime cost? Ans. 22s. 2ld. 20. Having sold 12 yards of cloth at 20^. per yard, and gained 10 per cent., what was the prime cost .'' Ans. 1 Si-. 2fjd. 21. Having sold 12 yards of cloth for £5 14^., and gained S per cent., what was the prime cost per yard.? Ans. 8,?. 9§r/. 22. For what did I buy 3 cwt. of sugar, which I sold for dE6 3a-., and lost 4 per ceait. } Ans. For £Q ^s. IJ-^. 23. For what did I buy 53 yards of cloth, which I sold for £25, and gained £b \0s. per cent. } Ans. For £23 135. 111(7. QUESTIONS. 1. What is the object of the rule .? [17]. 2. Given the prime cost and selling price, how is the profit or loss found } [17]. 3. How do we find the profit or loss per cent.? [18]. 4. Grivcn the prime cost and gain, how is the selling price found } [l-Jj. 5. Given the gain or loss per cent, and selling price, how do we find the cost price .? [20] . FELLOWSHIP. 21. This rule enables us, when two or more persona aie joined in partnership, to estimate the amount of profit or loss which belongs to the share of each. h'idlowship is either single (simple) or double (com- pound). It is single, or simple fellowship, when tlia diflerent stocks have been in trade for the same time. It is double, or compound fi-llowsliip, when the difiercnt ^stoi'ks luive biieu employed for diJJV.reiU times. This rule also enal)]es us to esti late how much of a bankrupt's stock is to ])e given to each creditor. 293 FELLOWSHIP. 22. Single Felloivship. — Rulr. — Say, as the wliolo stock^ is to the whole gain or loss, so is each pr-rson's contribution ^o the gain or loss which belongs to him. Example.— A put £720 into trade, B £340, and C ^eOGO ; and they gained Ml by the traffic. What is li'a share of it ? £ 720 .. 340 960 2020 : £47 :: £310 X47X340 — 2020~~ ^^* Each person's gain or loss must evidently be proportionai, to his contribution. EXERCISES. 1. B and C buy certain merchandizes, amounting to £80, of whicli 13 pays £30, and deSO ; and they gain £20. How is it to be divided .? Ans. B £7 10s , and £12 10.v. 2. B and C gain by trade £182 ; B put in £300, and £400. What is the gain of each t Ans. B £78, and C cii5l04. 3. 2 persons are to share £100 in the proportions Of 2 to B and 1 to C. What is the share of each > Am. B £66|, C £33|-. 4. A merchant failing, owes to B £500, and to £900; but has only £1100 to meet these demands. How much should each creditor receive ? Atis. B £3924, and C £707f ^' 5. Three merchants load a ship with butter; B gives 200 casks, C 300, and D 400 ; but when they are at sea it is found necessary to throAV 180 casks over- board. How much of this loss should fall to the share of each merchant ? Ans. 60, and D SO. 6. Three persons are to pay a tax of £100 accord- ing to their estates. B's yearly prapcrty is £800, G'a £600, and D's £400. How much is eacli person's share ? B should lose 40 casks, Ans. n\. is £44:^ C's £33^, and D's £223. 7. Divide 120 into throe sueh parts as shall be to each other as 1, 2, and 3 ? Ans. 20, 40, and GO. FELLOWSHIP. 293 S. A' ship worth £900 is entirely lost ; } of it be- Itmged to 13, J- to C, and the rest to D. What should be the loss of each, i3540 being received as insurance ? Ans. B £45, G £90, and D £225. 9. Three persons have gained £1320 ; if B were to take £6, C ought to take £4, and D £2. What is each person's share ? Ans. B's £660, C's £440. and D's £220. 10. B and C have gained £600 ; of this B is to have 10 per cent, more than C. How much will each receive .? Ans. B £314f , and C £2854. 11. Three merchants form a company; B puts in £150, and C £260 ; D's share of £62, which they gained, comes to £16. How much of the gain belongs to B, and how much to C ; and what is D's share of the stock ? Ans. B's profit is £16 165. 7j\d., C's £29 3s. 4^^d. ; and D put in £142 12s. 2^^c?. 12. Three persons join ; B and C put in a certain stock, and D puts in £1090 ; they gain £110, of which B takes £35, and C £29. How much did B and C put in ; and what is D's share of the gain ? Ans. B put in £829 Gs. ll^J^., C £687 3s. 5i-|J. ; and D's part of the profit is £46. 13. Three farmers hold a farm in common ; one pays £97 for his portion, another £79, and the third £100. The county cess on the farm amounts to £34 ; what is each person's share of it ? Ans. £11 18s. U^^d. ; £9 14s. 7^^d. ; and £12 6s. 4^^d. 23. Compound Fellowship. — Rule. — Multiply each person's stock by the time during which it has been in trade ; and say, as the sum of the products is to the whole gain or loss, so is each person's product to his share of the gain or loss. KxAMPLK.— A contributes £30 for 6 months, B £84 for 11 months, and C £9G for 8 months; and they lose £14. What is C's share of this loss 1 30 X 6=180 for one month. ) 84x11=924 for one month. } =£1872 for one month. y(3X 8=/U8 tor one mo nth. V 1872 : £14 : : £708 £14x708 "1.S72 _ =£0 Is. 4ld., C's bharo ir ; I Ji^i 294 FELLOVVSiJip. i ^„ ii iuunui , <inu, lor the same reason R'r no ^O'U for Uio same time; ami C's -m /"/«« oi =« ^ fu .^"* EXERCISES. in lio^'^i^""? '"5<^^^°t,s enter into partnership ; B puts S is. /n % ^ ?r'^'' ? ^'^ '^'- ^°r ^ months, Vu 1>^38 105 for 11 months; and they gain £86 16* j-^o lus., OS i.37 2s., and D's ^£24 4^ T^ in;«■^^^' '''"'^ ^-P'J ^^ ^' *'^^ :^^^^'« ^-^nt of a farm, and 1) 50 for the rest of the time. How much of the Ind D ^ii ^'''''' ^^^ ' ^'''- ^ ^^^ - ' ^ ^I^t't. and In ^^^'^^^?"^t':^' A' I^' «"<i C, enter into partnership, iAo wo"- fVT "^^^^-^SQl 13.. 4.Z.'^A's stock C's i 2^ '?« ''^'.^ "'^l.^rl' ?'^' ^200' 3 months ; and t s, X125, 16 months. What is each person's share of 131^47 '• ' '' '^^^' ^'' ^^0' ^^^ C'^ ^166 17. Three persons have received ^£665 interest- B ^nvf 1^ *t ? ?' ^ ''^''''^^^ 5 ^^^^ "^"«^ is each person's i)'si2oo ^''^' -^'^ ^^^^' ^'' ^^^^' ^^ trado* f'Zl^v^ '^I'^T ^«.°"ipa°y- X's stock is in trade 3 months and he claims J^ of the gain : Y's Btock IS 9 months in trade ; and Z advanced^e756 for 4 months, and claims half the profit. How much did X and Y contribute } Ans. X ^£168, and Y £280. It follows that Y's gain was A. Then -'- • » • • 4"T=.e.syA . pay it60 ; the first sent into it 56 liorses for 12 days, tho FELLOWSHIl'. 295 Bocond64 for 15 days, and tlio third SO for IS days. What must each pay ? Am. The first must pay £17 10s,, the second £2o, and the third i;37 10a-. 20. Three merchants are concerned in a steam vessel ; the first, A, puts in £240 for 6 months ; the second, ]J, a sum_ unknown for 12 months ; and the third, C, ^£160, for a time not known when the accounts were settled. A received £300 for his stock and profit, B £000 for his, and C £200 for liis ; what was B's stock, and O's time ? Ans. B's stock was £400 ; and C's time was 15 months. If £300 arise from £240 in C months, £000 (B's stock and profit) will bo found to arise from £400 (B's stock) in 12 months. Then £400 : £160 :: £200 (the profit on £400 'n 12 montlis) : £80 (the profit on £100 in 12 months). And £l604- 80 (£1G0 with its profit for 12 montlis) : £260 (£160 with Its profit for some other time) :: J2 (the number of months •^ *u s 260x12 , in the one case) : j^Xg^ (the number of months in the other casc)=]3, the number of months required to produce the difterence between £160, C's stock, and the £260, which he received. 21. In the foregoing question A's gain was £60 during (3 months, li's £200 during 12 months, and C'a £100 during 13 months; and the sum of the- products of their stocks and times is 8320. What wri(> their stocks ? Ans. A's was £240, B's £400, and C's £160. 22. In the same question the sum of the stocks is £800 ; A' stock was in trade 6 months, B's 12 months, and C's 15 months; and at the settling of accounts, A is paid £60 of the gain, B £200, and C £100. What was each person's stock ? Ans. A's was £240, B's £400, and C's £160. ' QUESTIOiS'S. 1. What is fellowship .? [21]. 2. What is the difterence between single and douhle fellowship ; and are those ever called })y any other names .^ [21]. 3. What are the rules for single, and double fellow- ship .? [22 and 23]. ' litmmi wSi il II 'l^H m 296 BARTKU. BAUTER. 25. Barter enables the merchant to exchange ono commodity for another, without either loss or gain. lluLE.— Find the price of the given quantity of ono kind of merchandise to be bartered ; and then ascertain how much of the other kuid tliis price ought to pux- . chase. ExA>[PT,E 1.— How much tea, at 8s. per lb, ou^ht to be t'-ivcn for 3 cwt. of tallow, at £1 10s. Sd. per cwt. 1 £. s. d. 1 16 8 3 5 10 is the price of 3 cwt. of tallow. And £5 10s.-^8s.=13^, is the number of pounds of tea which £o 10s., the price of the tallow, would purchase. There must be so many pounds of tea, as will be equal to the number of times that 8s. is contained in the price of tho tallow. E.vAMPi.E 2.— I desire to barter 96 lb of sugar, which cost me Sd. per lb, but which I sell at 13rf., giving 9 months' credit, for calico which another merchant sells for lid. per yard, giving months' credit. How much calico ouglit 1 to receive l I first find at what price I could sell my sugar, were I to give the same credit as he does — If 9 months give me 5d. profit, what ought 6 months to giveT 9 : 5 .6X5 30 ._gv/ 9 ~9~" ' ■ Hence, were I to give months' credit, I should charge ll»f/. per lb. Next— As my selling price is to my buying price, so ought his soiling to be to his buying price, both giving the same credit. lit : 8 :: 17 :5>^=12.Z. Tlie ]irico oi my f?ugar, inuroiuri;, is t?o a <-•"•> "'• '^^^•■•f md of his calico, 12r/. per yard. Hence "^^^=04, is tho required number of yards. BARTER 297 EXERCISES. 1 . A mevcliant lias 1200 stones of tallow, at 2s. 3ld. Iho Ktonc ; 13 has 110 tanned hides, weight 3994 lb, at b^d. the lb ; and thoy barter at these rates. How much ijwney is A to receive of li, along with the hides > Ans. £40 ll5. 2hL , , ,^ ^j 3. A has silk at Ms. per !b ; B has cloth at 12s. 6rf. which cost only 10s. the yard. How much must A charge for his silk, to make his profit equal to that of B ? Ans. 17s. 6d. 3. A has coffee which he barters at lO^Z. the lb more than it cost him, against tea which stands ]5 in lOs., but which he rates at 12s. Qd. per.tb. How much did the coffee cost at first ? Ans. 3s. 4d. 4. K and L barter. K has cloth worth 8s. the yard, which he barters at 9s. Sd. with L, for linen cloth at 3s. per yard, which is worth only 2s. 7d. Who has the advantage ; and how much linen does L give to K, for 70 yards of his cloth .? Ans. L gives K 215f yards ; and L has the advantage. f). 1) has five tons of butter, at £2o lOs. per ton, and lOi tons of tallow, at £33 15s. per ton, which he barters witli ; agreeing to receive i2150 Is. 6d. in ready money, and the rest in beef, at 21s. per barrel. How many barrels 's he to receive > Ans. 316. 6. I hi've cloth at Sd. the yard, and in barter charge for it at 13^/., and give 9 months' time for payment; mo,. ''^ant has goods which cost him 12^. per lb, an hich he gives 6 months' time for payment. IIow hi : he charge his goods to make an equal barter .^^ ...... At 17^^. ,^ , . . 7. I barter goods which cost 8d. per lb, but tor which I charge 13f^., giving 9 months' time, for goods which are charged at 17 d., and with which 6 months' time are given. Required the cost of what I receive > Ans. I2d. 8. Two persons barter ; A has sugar at Sd. per lb, charges it at 13d., and gives 9 months time ; B has at 12d. per lb, and charges it at 1 7d. per lb. How time must B give, to make the barter equal? 6 months. •f 298 ALLIGATION. QUESTIONS. 1. What is barter ? [25]. 2. Wiiat i.s tlie rule for k 'arter? [25], ALLIGATION. t IS called alligation medial; or what in-redients wl oe rcQuircd to np/w]i.«,> „ ^ • . ^^o^^uiLms will tlioy will produce— 'fertaicnts, to Iind the mixture nmnbc,- of tl,e lo«.o.st denomination confined in th^ whole ,,„a„„ty, „„d tho qnotient will boX Ate or d. d. 9X08 = 882 6x87 = 435 6x34 = 204 . 219 219)"l52l Ans. Id. per ib, nearly. The price of each e']"-nr. is fhp nnmV.«« „*• multipliod by tho iunn%;r if pou d ami fh^' ^'' P^"?^ whole is the mm of tho pricon B t 'if ° IQ ih /'"'"^ "^ ^^'° cost lo21./., ono 11.. ov the '^1o[i, !^A, .^^?.^^*'^^"8'^^' ^^'ive 21UtI part of thiH, li I piirt of ]621t/ or '-A^'</ ~ ; lust cost the ALLIGATION. 299 KxAMPLK 2.— What will bo tho price iinv II) of a mixtiiro ooiitainins !) lb G oz. of ten at 5s. Or/, per lb, 18 lb at (5> per lb, and 4() lb 3 oz. at U.s. 4^^/. per lb « lb oz. n. 9 6 at 5 18 G 46 3 9 d. £ s, 6 per lb= 2 11 per lb= 5 8 4iperlb=21 13 d. GJ 9 1177 )29 12 G ; Ans. 6d. per oz. nearly 73 IG Il77 ouncoa. And Gd. X 10=8.^., is the price per pound. In this case, tho lowest denomination beinff outice.M wo reduce the whole to ounces ; and having found the price of an ounce, wo multiply it by IG, to find that of a pound. E.YAMPi E 3.— A goldsmith has 3 lb of p;old 22 carats line, and 2 lb 21 carats lino. What will bo tho linoutss of tha mixture ? In this case the value of each kind of in-rrcdient is iT.n-c scnted by a number of carats — lbs 3x22 = GG 2x21 = 42 5 5)108 Tlie mixture is^^ carats fine. EXERCISES. 1. A vintner mixed 2 gallons of wine, at lis. por gallon, with 1 gallon at 124-., 2 gallons at 9^., and 4 gallons at 85. What is one gallon of tho mixture worth ? Ans. 10s. 2. 17 gallons of ale, at 9d. per gallon, 14 at 7i^., 5 at 91^/., and 21 at 4ir/., are mixed together. How much per gallon is the mixture worth ? Ans. 7j\d. 3. Having melted together 7 o?.. of gold 22^ carats fine, 121 oz. 21 carats fi'no, and 17 oz. 19 carats fine, I wish to know the fineness of each ounce of the mixture ? Ans. 20|f carats. 28. Alligation Alternafe. —Given the nature of the mixture, and of the ingredients, to find the relative amounts of the latter — ^ KuLE. — Put down the quantities greater than tho given mean (each of them connected with the differenco r • 300 AM.IOATION between it and the moan, by tlio Higii — ) in one column ; put tlio difforences botwcnn the remaining (luautitiea and the moan (eonncctcd with the quantities to which they belong, by the sign + ) in a column to the right hand of the former. Unite, by a line, amlipliis with souio viinus difference ; and then each difference will cxprii.ss how much of the quantity, with whoso difference it is connected, should be taken to form the required mixture. If any difference is connected with more than one other difference, it is to be considered as repeated for each of the differences with which it is connected ; and thef sum of the differences with which it is connected is to be taken as the required amount of the (Quantity whose difference it is. Example 1.— How many pounds of tea, at 5.5. and 8.?. per lb, would form a mixture worth 7."?, per tb '? Price. Diflerences. Price, i 1 S. S. The mean=8— 1- .V. s. -2-f-5=:thc moan. 1 IS connected with 2s., the difference l)otween the mean and 5s. ; hence there must bo 1 lb at 5s. 2 is connocled with 1, tlie difference between 8.<?. and the moan ; honco there must be 2 lb at 8s. Then 1 lb of tea at 5s. and 2 ib at 8.s-. per ib, will form a mixture worth 7s. per lb — as may bo proved by the last rule. It is evident that any equimultiples of these quantities would answer equally well ; hence a great number of answers may be given to such a question. Example 2.— How much sugar at Od, Id, 5d., and 10'/ , will produce sugar at 8>;/. per ib ? Prices. Hirt'eronces. Prices. The mean= d. d. 9-1- 10-2- d. d -3+5 the mean. 1 is connected with 1, the difference between Id. and the mean ; hence there is to be 1 ib of sugar at Id. per lb. 2 is connected with 3, the difference between 5d. and the mean ; hcwee there is to be 2 lb at M.. 1 is connected with 1, the difTerence between 9 J. and the mean ; hence there is to be 1 lb at 9f/. And 3 is connected with 2, the difference between lOf/. and the mean; hence there are to bo 3 lb at ]0c/. per ib. AI.MUAMON. 301 CoDHcrniontly wo nro to tiiko I lb ut 7>l., and 2 lb at 5</., 1 tb at ','*/., uud 3 11) ut lOil. If wo exuiuino wliat inixturo tUeso will give [27], wo Hhall find it to bo tlio givon moan. ExAMJM.K 3.— What quantities of tea at 4s., 6a., Ss. 0.«. por lb, will pi'oduoo aniixturo worth Ss. ? I'l-ices. Dift'ureucei. Tricui, S. S. 1 -f-4=tho mean. and Tho moan= 9-4 3, 1, and 4 aro connected with 1.?., the difforooo between 4<. juid tho mean ; thorofore wo aro to tako 3 lb -f- 1 lb -}- 4 lb of tea, at 4s. per lb. 1 ia connected with 3.'?., l.-j,, and 4.s'., tho ditferoncos between 8,s\, Gs., and 9s., and tho moan-, thoroforo wo aro to take 1 lb of tea at 8s., 1 tb of toa at Ga-., and I lb of tea at 9s. por lb. Wo Und in this oxampio that 8s., 6s., and 9s. aro all oon- nocted with the same 1 j this shows that 1 lb of oaoh will be required. 4s-. is oonnoctod with 3, 1, and 4; there nmsU bo, therefore, 3-f-l+4 lb of tea at 4s. ExAMPMc 4. — How much of anything, at 3s., 4s., 5s., 7?., 8s., 9s., lis., and 12s. por lb, would form a mixture worth Gs- per lb '? Pricci. Diirorences. Prlcei, Gs 1 lb at 3r, 2 lb at 4s., 3 lb at 7s., 2 lb at 8s., 3-|-5+6 (14) lb at Ss., 1 lb at 9s., 1 lb at Us., and 1 lb at 12s. per lb, will form tho required mixture. 29. Reason ok , the IIule. — The excess of one ingredient above the mean is made to counterbalance what the other wants of being equal to tho mean. Thus in example 1, 1 lb at 5s. per lb gives a deficiency of 2s. : but this is corrected by 2s. excess in the 2 lb at 8s. per lb. In example 2, 1 lb at Id. gives a deficiency of Id., 1 lb at 9^/. gives an exce.ss of Irf. ; but the excess of Id. and the deficiency of Id. exactly neutralize each other. Again, it is evident that 2 lb at 5.Z. and 8 lb at 10^. are Worth just as much as 6 lb at 8rf.— that is, Sd. will b« tha ftverugo price if w« mix 2 ib afc iiU. with '6 lb at lOd. 302 ALLIGATIOiV. ■, •»«» -#<i^ EXERCISES. 4. How much wine at 8s. 6d. and ds. per gallon will make a mixture worth 8s. lOd. per gallon.? Ans. 2 gallons at Ss. 6d., and 4 gallons at Qs. per gallon. 5. IIow much tea at 65. and at 3s. Sd. per Il>, will make a mixture worth 4s. Ad. per lb .? Am. 8 Sb at (is.y and 20 lb at 3s. 8^. per lb. 6. A merchant has sugar at 5r/., 10^., and I2d. per lb. How much of each kind, mixed together, wili be worth M. per lb .? Am. 6 lb at 5^., 3 lb at IQd., and 3 lb at I2d. ' 7. A merchant has sugar at bd., 10^., 12^., and 16'^ per lb. How many lb of each will form a mixture worth lU. per lb? Am. 5 lb at bd., 1 lb at 10^., 1 lb at 12(Z., and 6 lb at 16^. 8. A grocer has sugar at bd., Id., 12d., and 13d.. per K). ; How much of each kind will form a mixture worth lOd. per lb .? Am. 3 lb at 5d., 2 lb at 7d.,3fb at 12d.j and 5 lb at 13^. 30. When a given amount of the mixture is required, to find the corresponding amounts of the ingredients— Rule.— Find the amount of each ingredient by the last rule. ^ Then add the amounts together, and say, as their sum is to the amount of any one of them, so is the required quantity of the mixture to the correspondinff amount of that one. Example 1.— What must be the amount of tea at 4s. per ft, m 736 lb of a mixture worth 5s. per lb, and containing tea at bs., 8s., and l)s. per lb ? To produce a mixture worth 5s. per lb, we require 8 lb at 4s., 1 at 8s., 1 at 6s., and 1 at 9s. per lb. [28]. But all ot these, added together, will make 11 lb*, in which there are 8 lb at 4s. Therefore lb 8x736 tt> oz. =526 4y*y, the. required quantity lb 11 lb 8 m 736 11 of tea at 4s. That is. in 736 lb of flm mlx+nro thp"-" m. at 'rs. per lb. The amount of each of the other ingre dicnts may be found in the same way. Triix tJ\^ xju\J lU I,",- ALLIGATIOX. 303 rrnJi^l^ F 'u}'T' ^^"^ of Syvacu.o, £;ave a certain quaali y of p.ld o fom a ciwn; but when he received it, suHpectmo; that th« goLl.niith had taken son^e of the gold and «upp hed it« place by a b.ser metal, lie co,nmi«simied Auh .).ed_(^S the celebrated mathematician of Syracuse, to TZ'T^'x "' '^fP'^i'^"/^^^^^ ^vell founded, aril to what oxtrnt Archiriiedes was tor some time unsuccessful in his resoarches, unti one day, goin- iuto a bath, he rcmark.Kl that he displaced a quantity of water equal to his own bulk Seeing at once that the same weight of different bodies wou.d, Jf "nmcr,sod in water, displace very dilFcrent quan- tities of the fluid he exclaimed with delight that he had found the desired solution of the problem Taking a mass j>t guld equal HI weight to whatwa^ given to the gohlsmith, he tound thnt it displaced less water than the crown : which t.ieretoro, was made of a lighter, becnn.so a more bulky mortal— and, consequently, was an alloij of goM - jNow supposing copper to have boen the substance with wliich the crown was adulterated, to find its amount- J.et the goiu given by Hiero have ww£rhed 1 lb, this won d displace about -O.IL' lb of water; 1 lb ^.f copper void d.sp ace about -1124 3b of Avater; but let the criwn have displaced only -072 it). Then "^ uo\mi .ia\e (rold differs from -072, the memi, by— •020 Copper differs from it by . . -f-O-lO-l", ,T ., Copper. Di'I'LTPiices. (Jold. Hence, ths moan=.=. 1124 -0404 •020-f-052=thc mean. Therefore -020 lb of copper and -0404 ib of gold would t>r()duce the alloy in the crown. ^ l>ut the crown was supposed to weigh 1 ib ; therefore •0G04 lb (-020+ -0404) : -0404 lb • • lib • li^Mil"* •0G04 •GG9=-331 lb is GC9 ib,_ the quantity of gold. And 1- thc quantity of coppc EXERCISKS. U. A di-usrgist IS desirous of producinir, from medicine at '>J'., (^'.v., S.v and 9.-. per 3b, li cwt. of a mixture worth 7s per ]b. How much of each kind must he use for the purpose r Ans. 28 lb at 5.$., 56 lb at 6s., ufv xKf ai, 05., ana 2:i ih at bi'. per ib. 10. 27 lb of a mixture worth 4s. 4d. per Ib are re- qiured. It IS to contain tea at 5^. and at 3s. 6d. per 304 ALLIGATION. lb. How mucli of each must bo used ? Ans. 15 ft) at 5i-., and 12 ib at 3.9. 6cL 11. How much sugar, at Ad., Gd., and Sd. per ]b, must there be in 1 cwt. of a mixture worth 7d. per ib } Ans. 18|- lb at 4(/., ISf lb at 6d., and 74| lb at 8d. per lb. 12. How much brandy at 123., 135., 145. , and 14a'. Gd. per gallon, must there be in one hogshead of a mix- ture worth 135. Gd. per gallon > Ans. 18 gals, at 125., 9 gals, at 135., 9 gals, at 145., and 27 gals, at 145. Gd. per gallon. 31. When the amount of one ingredient is given, to find that of any other — lluLE. — Say, as the amount of one ingredient (found by the rule) is to the^ii-m amount of the same ingredient, so is the amount of any other ingredient (found by the rule) to the required quantity of "that other. Example 1.— 29 lb of tea at As. per lb i« to bo mixed with teas at (js., 85., and Ds. per lb, so as to produce what will be •Vortli 5.s\ per lb. What quantities must be used '? 8 Ih of tea at 45., and 1 ib at 6s., 1 lb at 8s., and 1 lb at 9s., will make a mixture worth 5s. per lb [271. Therefore 8 ib (the quantity of tea at 4s. per Ib, as found by the rule) . 29 R) (the given quantity of the same tea) : : 1 lb (the quantity of tea at Gs. per ib, as found by the rule) : 221^ ^^ 8 rthe quantity of tea at 6s., Avhich corresponds with 29 lb at 4s. per lb) ==3-^ lb. We may in the same manner find what quantities of tea nt 8s. and 9s. per lb correspond with 29 lb— or i\\Q given amount of tea at 4.s. per lb. Example 2.— A refiner has 10 ov.. of gold 20 carats fine and melts it with 16 oz. 18 carats fine. What must be added to make the mixture 22 carais fine ? 10 oz. of 20 carats fine=10x20 = 200 carats. 16 oz. of 18 caratii fme=16xl8 = 288 26 : 1 : : 488 : 18}'[ carats, the fineness of the mixture. 24 — 22=2 carats baser metal, in a mixture 22 carats fine. 24 — 18f|=5j% carats baser metal, in a mixture 18 JH carats fine. Then 2 carats : 22 carats : : 5^^^ : 57 j''^ carats of pure ALLIOATION. 305 fro] (1- required to ohanse 5 ■',- carats baser metal, into a mixture 22 carats line. Ikit tliero are already in the mixtura 1S|:; Ciirats gcl.l; therefore 57^^,— 18j!;:=:i8f!| carats ir<M are to, l)e added to every ounce. There are 20 oz.; therefore 2GXoH|.;=1008 carats of gold are wanting. There are L4 carats ^( page 5) in^everyoz. ; therefore 'i;^^ caratsrr^-12 '" """'" ' ' " ' There will then' he a uiixturo oz. of gold must l)e added containing oz. car. 10X20 ]()Xl8 42x24 car. 2')0 288 1008 08 : 1 oz. : : 14DG : 22 carats, the required finoness. EXERCISES. 13. How iTiiicli tea at 6s. per lb must be tnixod with 12 ii) at 3i-. ikl. per il), so that the mixture maybe worth •].?. 4d. per lb .? Ans. 4f lb. 14. How much brass, at I4d. per tb, and pewter, at lO^d. per lb, must I melt with 50 lb of copper, at 16V/. per lb, so as to make the mixture worth Is. per lb ? Ans. 50 lb of bra.s.s, and 200 lb of pewter. 15. How murdi gold of 21 and 23 carats fine must be mixed with 30 oz. of 20 carats fine, so that the mix- ture uuiy bo 22 carats fine r A)is. 30 of 21, and 90 of 23. 16. How much wine at 7s. r^d.^ at 5.?. 2d., and at 4s. 2d. per gallon, must be miyxMl with 20 gallons at O.v. 8^/. per gallon, to make the mixture worth 6s. per gallon r Ans. 44 gallons at 7s. ixL, 16 gallons at oa- 2d., and 34 gallons at 4i-. 2d. QUESTIONS. 1. What is alligation medial .? [26]. 2. What is th,^ rule for alligation me lial > [27]. 3. What is alligjition altoniato : [26 K 4. Whnt is the rule for Jilligatim alternate } [28]. 5. What is the rule, v.hon a certain amount of t) <j mixture is required .? [30] . 6. AVhat is the rule, when i\\(^. a; m\\\ 0\ C'l* or moro of the ingredients is <^iveu .^ [31]. 306 SECTION IX. INVOr.UTION AND EVOLUTION, kc. 1. iNVOLUxroN. — A qnantlty wliicli is the product of two or more factors, each of theiu llie same number, is termed a power of that number ; and the number, mul- tiplied by itself, is said to })0 invclccd. Thus SXoXo (:-^125) is a " power of 5 ;" and 125, is 5 " hivolved." A power obtains its denomination from the number of times the root (or quantity involved) Is taken as a factor. Thus 25 (=5X5) is tlie secovd power of 5. — Tlie second power of any number is also called its square. ; because a square surface, one of M'hose sid' s is expressed by the given number, will have its area indicated by the second power of that nun ber ; thus a square, 5 inches every way, will contain 25 (the S(|uare of 5) square inches ; a Sfjuare 5 feet (svery way, will contain 25 .square foot, &c. 216 (6X0X<)) is "the lliird power of 6. — The third power of any nundjer is also termed its mill ; because a cube, the length of one of avIi ^e sides is expressed by the given number, will have ils solid contents indicated by the third power e.f that number. Thus a cube 6 inches every way, will contain 125 (the cube of 5) cubic, or solid inches; a cube 5 feet every way, will contain 125 cubic feet, tic. 2. In place of setting dov/n all the factors, we put down only one of them, and mark how often they are supposed to be set down by a small figure, which, since it poin/s out the number of the factors, is called the i7idc.x, or cxpinnanf:. Tlius ^^ is the abbreviation for 5x5 : — and 2 is th>5 index. 5^ moans 5X5X5X5X5, or 5 in the fifth power S"* means 3X3X3X3, or 3 in the fourth power. S' moans 8X8X8X8X^X8X8, or 8 in the seventh power, &c. 3. Someti)nes the vinculum [See. IT. 5] is used in con- junction with the index ; thus 5-f-'82 means that the sum of 5 and 8 is to be raised to the second power — this INTOLUTION. 307 is very ciIiTerent from 5 ^+8 ° , wlncli means tlic sum of the squares of 5 and S : 5 + 8= being 169 ; while 5^ + S'' is only 89. 4. Iq multiplication the multiplier may be considered as a species of index. Thus in 187x5, 5 points out how often 187 should be set down as an addend ; and 187X5 is merely an abbreviation for 187+187+187 + 187+187 [Sec. 11.41]. In 187% 5 points out how often 187 should be set down as a factor ; and 187* ig an abbreviation for 187X 187X 187X 187x 187 :— that is, the " multiplier" tells the number of the addends^ and the " index" or " exponent," the number of the factors. 5. To raise a number to any power — Rule. — Find the product of so many factors as the index of the proposed power contains units — each of the factors being the number which is to be involved. Example 1. — What is the 5th power of 7 "? 7» =7x7x7x7x7=10807. Example 2. — What is the amount of £1 afc compound interest, for 6 years, allowing G per cent, per annum 1 The amount of XI for G years, at 6 per cent, is — _10GxlOGxl-OGxl-06xlOGxl-06 [Sec. VII. 20], or 1-00"=1-41852. We, as already mentioned [Sec. VII. 23], may abridge <\.o process, by using one or more of the products, already obtained, as factors. ■^ EXERCISES. 1. 3'=243. 2. 20'"=I0240000000000. 3. 3^=2187. 4. 105''=1340095r>40r)25. 5. 105''=l-3400956-10G25. ^6. To raise a fraction to any power — ' Rule. — Raise both numerator and denominator to that power. Example. — (f)=^ to maliiply it ny itself. But to multiply it by itself any nuuil)er of tiinos, we must multiply its numerator by itwelf, and also its deuomiuator by itself, ihaf number of times [Sec. IV. 00]. ^os EVOLUTION. R /•.■^^7 'J Mil "• U ^ — 11T38V - () / o\n :i I :.';■. 7. To raise a mixed nunibor to any power — lluLE. — llcduoo it to an improper fraction [Sec. IV 24] ; and then proceed as directed by the lust rple. EXAMI'LK.— (21)4=(|)4=fyi^5. EXERCISKS. 10- K-r.J - T.^ 11. (3^)^=«u^;;^^ - - ■■ - .2 2 1 (i-ir £9 8. Evolution is a process exactly opposite to mvolution , since, by means of it, v/e find what number, raised to a given power, would produce a given quantity — the num- ber so fnund is termed a root. Thus wc " evolve " 25 when wo take, for instance, its square root ; that is, wh-en wc find what number, multiplied by itself, will produce 25, Roots, also, are expressed by e.rjjonenls — but as these exponents are fractions, the roots are called ^^ fractional powers." Thus 4^ means the square root of 4 ; 4^ the cube root of 4 ; and 4^ tlie seventh root of the fifth power of 4. Hoots are also expressed by ^, called the radical sign. When used alone, it means the square root — thus ^3, is the square root of 3 ; but other roots are indicated by a small figure placed within it — thus ^6 ; which means the cube root of 5. ^7^ (7^)? is the cube root of the square of 7. 9. The fractional exponent, and radical sign are some- times used in conjunction with the vinculum. Thua 4—3% is the s quare root of the difierence between 4 and 3 ; ^o-{-7^ or 5+7'^, is the cube root of the sura of 5 and 7. iO. To find the square root of any number — Rule — I. Point off the digits in pairs, by dots ; put- ting one dot over the units' jo/acfi, and then another dot over every second digit both to tha right and left of the units' place — if there are digits at both sides of the decimal point. EVOLUTION. 309 IT. Find the highest immber the square of which will not exceed the amount of the highest period, or that which is at tlie- extreuio hift— this number will bo the first digit in tho required square root. Subtract its square from the highest period, and to the remainder, considered as hundreds, add the next period. III. Find the highest digit, wliich being multiplied into twice the part of the root already found (consi- dered as so many tens) , and into itself, tho sum of tho products will not exceed the s^tm. of the last remainder and tho period added to it. Put this digit in the root after the one last found, and subtract the former si07>i from the latter. IV. To the remainder, last obtained, bring down another period, and proceed as before. Continue this process until the exact square root, or a sufiicicntly noar approximation to it is obtained. 11. I'LxAMPLK.— What is the square root of 22420225 '^ 22420225(4735, is the required root. 1G__ 87)042 ,-;;■ :"'•■ - GOO ,,. . . . • 943)3302 ■ "^-^ ' • .-'■ 2820 .-.-■■■ .. • 0405)47325 47325 22 i« tlie highest period; and 4^ is the highest square wlucli doo.s not exceed it— we put 4 in tlie root, and subtract 4'-', or 10 from 22. This leaves 0, which, along with 42, the next poriod. malccs 042. We subtract 87 (twice 4 tcns-{-7, the highest digit yhicIi wo can now put in the root) X 7 from 042. This loaves 33, which, along witli 02, the next period, makes 3302. We subtract 043 (twice 47 tens -\-'i, the next digit of th(^ root) X3 from 3302. This lca.vo.s 473, ^^-jiich, ixhmu V;:t'.^ 25, the only remaining period, makes 47325, We subtract 0405 (twico 473 tons J..'",, the np:.c digit of the root) X5. Thi.s leaves n-'^ romaiuder, The given numbp,v, therefore, is exactly r. square; and its squi\re root is 4735, 12. llKAsoiv OF I.— Wc point off ';„o .^ip-itg of tlie given square in pairs, and consider tlio ^^j,i^{)cr of dots as indicating 310 iJVOLUTIOr* ^^ii^ir^ ' "*■ ''■'='" '■" .""' '■'>"'• ''"'" ""M-or one nor two the root—since it will be necessarv fn K^,-^~\5 ^ '^ .^ , for each new digit; but Zr"o1hri^e*"w^rn«„?°b:?e°,K™'' Keabon or II —We subtract from llie Wcliost Mrio,l of '(!,. fml 00?^"'°''.,""' '■'8''''" 'I""' "W«h £ nJt ™co°d U dit of I „"■ "'? T' "' *■' "1""° »« the 8rst or hTlc,; 600 m JZ\, 'i?.'^"""^ by to digits mi. for iustance,"nto will contain not only IO2 and 42 hnf nian +!;• li ^^^"^^^ rvf in oTiri .1 \u "l/ , , * ' "^^ *'So twice the product £Xf ^r r s .t. rsriort^^i'triai cedS it "WX'Z^ " ''^ "!° "«' »f tl.e root wl^iSp? .' whenwesubracfS7v7 """Pf "''f'' "'•«*■■"*« the rule. 4000 =16000000 ' 6420225 2X4000X700+700^= 6090000 2X4000X30+2X700X30+30*= 282900 8X4000X5+2X700X5+2X30X5+5^=17325 EVOLUTION 3tl of twice tlic sum) of tlio parta of tlio mot nlrc.idy found, jnulMphed by tlio ncAV digit, Tims 22420225, the 8quavo of 4785 contains 4000^-f700--|-30^-f5^ and also Uvica 4000X 700 + twice 4000X30 4- twice 4000x5; plus twice 700x304- twice /00X6; phis twice 80x5:— that is. the square of each ot Its parts, with the euui of twice tlio product of every two of them (which is the same as each of tliem multiplied by twice the sum of all the rest). This would, on examination, be lound the case with the square e)f any other number. If we examine the cxamjile given, we shall find that it will not be necessary to bring down more than one period at a time, nor to add cyphers to tlie quantities subtracted. 13. When the given square contains decimals — ; If any of the periods consist of decimals, the digits m the root obtained on bringing down these periods to the remainders will also be decimals. Thus, taking the example jus t given, bu t altering th e decimal point, wo Bh all have ^2 24202-25= 473- 5; V224 2-0225=47-35. ^22-42022 5 = 4-735; V^2420225 = -4735 ; and ^•0022420225 = -04735, &c. : this is obvious. If there is an odd number of decimal places in the power, it must be made even by the add ition of a cypher. Using the same figures, ^2242022-5= 1497-338, &c. 2242022-56 (1497- 338, &o 24)124 _%_ 289)2820 2CM_ 2987)2H)22 20909 29943)101350 89829 299463)1152100 898389 _ 2994668)26371100 23957344 1413756 in this case the highest period consists but of a single digit nilU flip frlVf>n linTYlVvisV lO »irif o ^n-,<fnn4- <./>,,.'!«» There must be an even number of decimal places ; .lince nc number of decimals in the root will produce an odd numbe? in thi^ square [Sec. II. 48]— as may be proved by experimen* ia_Ji 312 EVOH KXKR JTION. CISES. 20. 21. 22. 23. 24. 25. 14. 15. 10. 17. yi95304=442 ^328329— 573 ^•0070= -26 ^87 -05=9 -3022 ^^801=29 -3428 ^984004=992 ^5=2- 23007 y- 6= -707 100 V'Ol -9081— 9-59 .y 238 144=488 18. 10. ^^^2 -3761=5 -09 ^•33 1770= -576 14. To extract the square root of a fraction — EuLE. — Having reduced the fraction to its lowest torni.s, make the square root of its numerator the nume- rator, and the square root of its donominatcr the deno- minator of the required root. Example.— y*=f. 16. Reason of the Rule.— The square root of any quau- tity must bo such a number as. multiplied by itself, will pro- duce that quantity. Therefore f^ is the square root of | ; for I y^ l=ff- ^^e same might be shown by any other example. Basides, to square a fraction, we must multiply its numera- tor by itself, and its denominator by itself [6] ; therefore, to take its square root— that is, to bring back both numerator and denominator to what they were before— we must take tbe square root of each. 16. Or, when the numerator and denominator are not squares — Rule. — Multiply the numerator and denominator together ; then make the square root of the product the numerator of the require 1 root, and the given denomi- nator its denominator ; or make the square root of the product the denominator of the requu'ed root, and the given numerator its numerator. Example.— What is the square root of f ? (|)J a =4-472136-{-5='894427. ^/1X5 ^^ 6 ^6X4 17. We, in this case, only multiply the numerator and denominator by the same number, and then extract the square root of each product. ^^^^ 5=5"^' or ^. Therefore (|)^ ''4x4 -a 4 V5X5/ 2 _-s/^X5 —- L--, or \5X4/ V5X4* A> EVOLUTION. 313 Ifi. Or, lastly— lliTLE. — Hediico the given fraction to a decimal [Seo IV. 63J, and extract its square root [13J EXERCIHKS. 20 /22\i 28-5300852 27 28. \37/ "^ 37 14 14 '9000295 6-244998 13/ 13 29. 30. Sli (^|-)^=-745350 (j^y.=:' 8000254 (f)'- 8451542 19. To extract the square root of a mixed number — Rule. — lleduce it to an improper fraction, and then proceed as already directed [14, &c.] Example.— y2.I=y^ =^=H. EXERCISES. 32. y51|j=71 33. y27VV=5i 34. yl ''o^lOlSSS 35. v'lI|=-l-lG83 3G. y_0,^=2-5298 37. ^'13^=30332 20. To find the cube root of any ni;..\ber — Rule — I. Point oif the digits in threes, by dots — putting the first dot over the units' place., and then proceeding boi/i to the right and left hand, if there aro digits at both sides of the decimal point. II. Find the highest digit whose cube will not ex- ceed the highest period, or that which is to the left hanu side — this will be the highest digit of the required root; subtract its cube, and bring down the next period to the remainder. HE. Eind the highest digit, which, being multiplied by 300 times the square of that part of the root, already found — being squared and then multiplied by 30 times the part of the root already found — and being multiplied by its own square — the su7)i of all the pro- ducts will not exceed the suvi of the last remainder and the period brought down to it. — Put this digit in tho root after what is already there, and subtract the former ium from the latter. IV. To what now remains, bring down the next r ■' 314 EVOLUTION. period, niul procooJ ns botorc. Continue tliia process until tlio exact cube root, or a suflicieutly near ajtproxU ination to it, is obtained. ExAMi'LE.--\Vhat i8 the cube root of 1795970G9288 ? 179597009288(5042, tho required root. 125 300x5»x0 30x5 xO» G'XO 30()x50»x4 30x50x4» 4«x4 300x5G4'''x2 30x504x2^ 2*x2 545!) = 500 3981009 3790144 190925288 190925288 We find (by trial) tliat 5 is tho first, the second, 4 tho third, and 2 tlio last digit of tho root. And the given number is exactly a cube. 21. IIeason of I. — We point off the digits in threes, for a reason similar to that which caused us to point thorn off in tf?os, when extracting the square root [12]. Reason of II. — Each cube will be found to contain the cube of each part of its cube root. Reasoist of III. — The cube of a number divided into any two parts, will be found to contaiu, besides the sum of the cubes of its parts, tlie sum of 3 times the product of «ach part by tlie otl.er part, and 3 times the product of each vart by the squaio of tho other part. This will appear from the following : — 179597069288 5000*=1 25000000000 54597069288 X 5000"' X GOO-f 3 X 5000 X G00'-}-G00*= 5061 6000000 3 X 5000' X 40-f 3 X 5G00 X 40^+40' 3981009288 3790144000 190925288 8 X 5640^ X 2 -(-8 X 5640 X 2*+2'= 1 90925288 Hence, to find the second digit of the root, we must find by tnai some rrarnbcr which — being multiplied hj 3 times the square of the part of the root already found — its square being EVOLUTION. 315 mnltiplio'l l»y H Hmoa tlio part of tho root nlromly fotin.l— and lii'itig iimltiplicd by tho nqunrc of UhoU'— tho Htim of the pro- ducts will not exceed wliat rornnins of tho j^fiven numhnr. JiiHtoiuI of couHideriiif^ tlio part of tlio rnot iilretidy fdund ns to many tens [i2J of the denoiuiiiatiou next fdllowiiig (jih it rt'iilly Ih), which woidd (idd one cypher to it, ami two cyphers to ItH aquaro, wo consider it as so many iinitH, and multiply It, not )>y 3, but by HO, ami its Bquarc, not by %, but by 800. For 800 X 5' X t5 -i- '"'^ X & X G'-f-G'X') Ih the sanio thing as 8xr>0'XG-f-3x50Xt»'+<)'X'»; since Ave only change tho posi- tion of the factora 100 and 10, which docs not alter tho product [Sect. 11. 35]. It in evidently unncccHsnry to bring* down more than ono period at a time ; or to add cypherB to tlie subtraliendH. Ukasov ok IV. — The portion of tlie root already fniind may be treated as if it "ro a sinfflo digit. 8inco into wliatever two parts wo («livido any number, its cube root will contain tlio cube of ench part, with o times the flquaro of each multi- plied into the other. 22. Whon there me. decimals in tho given cube — If any of the periods consist of decimals, it is evident that the difji;its found on bringinjjj down tbeso periods Ljust be decimals. Thus ^17U.5'}7-()6928S = 5n-42, &c. When the dtMJimals do not form complete periods, the periods are to bo completed by the addition of cyphers. ExABiPLE. — What is tho cube root of -3 '? 0'800(-CG9, &c. 21G 800X6'X6 SOXGXG^ GXG' 800 X 66' X 9 80XGGX9' 0X9* •669, &c. And 84000 =71496 12504000 =11922309 581G91, &c. ^•3='669, &c. And -3 is not exactly a cube. It is ncce.ssary, in this case, to add cyphers; since ono decimal in the root will give 3 decimal places in the cube; two decimal ^laces in the root will give six in the cube, &c. [Sec. II. 48.] KXKRCISES. 88. yp=3- 207534 89. 4/39=3 -391211 40. y2r2=5-962731 41 . ^n 23505'.!92=4 98 42. ^190r0U37"5=575 43. ;/458ai4011=771 44. ^ 483 • 736 (325=^7 -85 45. ^•G3G05a=-86 4(^ 3/099=') •ODGGGG 47. y- 979140657= -993 i! 1 i 1 31G EVOLUTION. .2 23. To extract the cube root of a fraction — 3luLE. — JIaving reduced the giveu fraction to its lowest terms, make tlie cube root of its numerator the luimerator of the required fraction, and the cube root of its denominator, the deuomiuator. ^''' ^125 21 Reason- of the Rule.— The cube root of any number must be such as that, taken three times as a factor, it will procluce that number. Tlierelbre f is the cube root of - 3^^; fov j X I X f = yI ^.— Tlie same thing might be shown, by uuv otiier example. '' Resides, to cube a fraction, we must cube both numerator au<? denominator; therefore, to take its cube root— tliat is to reduce It to what it was before— wo must take the cube root of both. 25. Or, when the numerator and denominator are not cubes — llui.E. — IMuItiply the numerator by the square of tlu^ denominator ; and then divide the cube root of the pro- duct by the given denominator; or divide tlie given numerator by the cube root of the product of the given denominator multiplied by the square of the giveu numerator. Example.— What is the cube root of 5 ? ^— 2 or -.^. = 5-277032 -^ 7 == 753047. (./ = ^3XP -5/7x3' This vale depends on a principle already explained [IG]. 26. Or, lastly— Rule. — lloduce the given fraction to a decimal [Sec. IV. 63], and extract its cube root [22] . 48. 40. 50. 8-G5349( \11/ ~5- 604079 EXEnCISE.S. 61. 52. 7-(>51725 (|^y=-560907 ■472103 27. To fijid the cube root of a mixed number — lluLE. — Iteduce it to :m improper fraction ; and then proceed as already directed [i>3, &c.] EVOLUTION. 317 EXERCISES. 54. (28ni=3-0G35 55. (7})J=l-93098 56. (9^)i=20928 57. (71f)*=41553 58. (32/y)^=31987 59. (5|)Ul-7592 28. To extract any root wliatever — liuLE. — When the index of the root is some power of 2, extract the square root, when it is some power of 3, extract the cube root o* the given number so many times, Buccessively, as that power of 2, or 3 contains unity. / Example 1.— The 8th root of 65530=>/Vy 65536=4, Since 8 is the third power of 2, we are to extract the square root three times, successively. Example 2.— 134217728«=yVl342lT7S=8. Since 9 is the second power of 3, we are to extract the cube root twice, suocessively. 29. In other cases we may use the following (Hutton Mathemat. Diet. vol. i. p. 135). Rule. — Find, by trial, some number which, raised to the power indicated by the index of the given root, will not be far from the given number. Then say, as one less than the index of the root, multiplied by the given number — plus one more than the index of the root, multiplied by the assumed number raised to tlie power expressed by the index of the root : one more than the index of the root, multiplied by the given number — plus one less than the index of the root, multiplied by the assumed number raised to the power indicated by the index of the root, : : the assumed root : a ^ still nearer approximation. Treat the fourth proportional thus obtained in the same way as the assumed number was treated, and a still nearer approximation will be found. Proceed thus until an approximation as near as desirable is discovered. Example.— Wliat is the 13th root of 923 1 Let 2 bo the assumed root, and the proportion will be 12x923+14x2^-' : 14x923+12x2*^ :: 2 : a nearer approximation. Substituting this nearer approximation for 2, in the above proportion, we get another approximation, which wo may treat iu the same way. 318 EVOLUTION EXKRCISK3. GO, (9GG98)K=G-7749 Gl. (GG457)iT=27-l42 62. (23G5)?=31-585 68. (8742G)?=5084-29 04. (8-9G5)'=l-368 65. (•07542G)t4=-04G988 30. To find the squares and cubes, the square and cube roots of numbers, by means of the table at the end of the treatise — This table contains the squares and cubes, the square Rnd cube roots of all numbers which do not exceed 1000 hut it will be found of considerable utility even when very hi£':h numbers are concerned — provided the pupil bears in^inind that [12] the square of ai\y number is equal to the sum of the squares of its parts (which may be found by the table) plus twice the product of each part by the sura of all the others ; and that [2 1 ] the cube of a number divided into any two parts is equal to the sum of the cubes of its parts (which may be found by the table) plus three times tne product of each part multi- plied by the square (found by means of the table) of Hie other. One or two illustrations will render this sufficiently clear. Example 1. — Find the square of S734r)G. 873450 maybe divided into two parts, 873 (thousand) and 45G (units) . But we find by the table that 873'=7G2120 and 450'=20793G. Therefore 762129000000=873000' 700176000=873000 X twice 45G 207936=450' And 702025383936=873456' ExAMPLK 2.— Find the cube of 864379. Dividint; this into 864 (thousand )_and 379 (units), wejfind 86?=(vi4972544 b'64 =746496, 379 =54439939, and 379 =143641 Therefore 644972544000000000=8(HOOO' 848765952000000=3 X 804 W X 379 3723 1 7472000=3 x 804000 x 3?j' 54439931 379 And G45821G82323911939=r-86^ LOGARITHMS. 319 <5l In finding tlie square and cube roots of larger numbers, we obtain their three highest digits at once, if we look in the table for the Jiighest cube or square, the highest period of which (the required cyphers being added) does not exceed the hio-hest period of the given number. The remainder of the process, also, may often be greatly abbreviated by means of ithe table. • QUESTIONS. 1. What are involution and evolution } [1]. 2. What are a power, index, and exponent > [1 & 2J. 3. What is the meaning of square and cube, of the B(iuare and cube roots } [I and 8J. 4. What is the difference between an integral and a fractional index .? [2 and 8] . 5. How is a number raised to any power } [5]. 6. What is the rule for finding the square root } [10]. 7. What is the rule for finding the cube root ? [20] . 8. How is the square or cube root of a fraction or of a mixed number found > [14, &c., 19, 23, &c., 27]. 9. How is any root found } [28 and 29] . 10. How are the squares and cubes, the square roots and cube roots, of numbers found, by the table .? [30] . LOGARTIHMS. 32. Logarithms are a set of artificial numbers, which reprcsent°the ordinary or 'natural numbers. Taken along with what is called the base of the system to which they belong, they are the equals of the corres- ponding natural numbers, but without it, they are merely their representatives. Since the base is un- changeable, it is not written along with tlie logarithm. The logarithm of any number is that power of the base which ts equni to it. Thus 10^ is eqital to 100 ; 10 is the hase^ 2 (the index) is the logarithm^ and 100 is the corresponding natural number.— Logarithms, therefore, are merely the indices which designate certain powers of some base. 33.. Logarithms afford peculiar facilities for calcu- lation. For, as we shall sec presently, the multiplica- tion of numbers is performed by the addition of their 320 LOGARITHMS. logarithms ; one number is divided by another if we subtract the logarithm of the divisor from that of the dividend ; numbers are iuvolveu J' we multiply tJioir logarithms by the index of the proposed power ; and evolved if wo divide their logarithms by the index of tho proposed root.— But it is evident that addition and subtraction are much easier than multiplication and division ; and that multiplication and division (particu- larly when the multipliers and divisors are very small) are much easier than involution and evolution. 34. To use the properties of logarithms, they must bo exponents of the same base— that is, the quantities raised to those powers which they indicate must be the same. Ihus 104X123 is neither 10^ nor 12% the former bein-. too small, the latter too great. If, therefore, we desirS to multiply 104 and 12« by means of indim, we must Imd .some power of 10 which will be equal to 123 or some power of 12 which will be equal to 10% or finally two powers of some other number which will be equal respectively to 10^ and 123, ^^^ then, adding these powers of the same number, we shall have that power ot It which will represent the product of 10^ and 123 Ihis explains the necessity for a table of loo-arithms— we are obliged to find the powers of some one base which will be either equal to all possible numbers, or so neariy equal that the inaccuracy is not deserving of notice The base of the ordinary system is 19 ; but it is clear that tiiere may be as many difierent systems of logarithms as there are difi-erent bases, that is, as there are difierent numbers. 35. In the ordinary system— which has been calcu- lated with great care, and with enormous labour, 1 is the logarithm of 10 ; 2 that of 100 ; 3 that of 1000, &c And, since to divide numbers by mejins of these loga- nthms (as wo shall find presently), we are to subtract the logarithm of the divisor from that of the dividend, IS the logarithm of 1, for 1=L^— 10'-'— 10" • — 1 Ls 10 ' the logarithm of -1, for •l=si=lo''=10«— 1^=^10- 10 101 for the same reason, -2 is the logarithm of -01 : that ol -001, &c. ' -1 . „„.! , uuu bcr, a LOOAUITHMS. 521 ^■', or '3 36. The logai-itlinis of numbers hetwem 1 must 'be more tlifin and less than 1 ; that is, Bomo decimal. and 1 0, , Kn.iu xn, must bo The logarithms of numbers between JO and 100 must be more than 1, and less than 2 ; that is, unity with some decimal, &c. ; and the logarithms of numbers between -1 and -Ql must be —1 and"' some deci- mal ; between -01 and -001, —2 and some decimal, &c. The decimal part of a logarithm is ahoays positive. 37. As the integral part or charaderisik of a posi- tive logarithm is so easily found — being [35] one less than tlie number of integers in its corresponding num* bcr, and of a negative logarithm one more than thu number of cyphers prefixed in its natural nuniber, it is not set down in the tables. Thus the logarithm corresponding to the digits 9872 (that is, its decimal part) is 99440:'^ ; hence, the logarithm of 9872 is 3 •994405 ; that of 9S7-2 is 2-994405 ; that of 9-872 is 0-994405 ; that of -9872 is- 1-994405 (since there is no integer, nor prefixed cypher) ; of -009872— 3*994405, &c. : — The same digits, whatever may be their value, have i\\Q E-AmQ decimals in their logarithms; since it is the integral part, only, which changes. Thus the logarithm of 57864000 is 7-702408 ; that of 57864, is 4-7G2408 ; and that of -0000057864, is— 6-762408. 38. To find the logarithm of a given number, by the table — Tlie integral part, or charaGtoristic, of the logarithm may be found at once, fioni v/liat has been just said [37] — When the number is not greater than 100, it will bo found in tlie column at the top of which is N, and the decimal part of its logarithm iinmediately opposite to it in the next column to tlio light liand. If the number is greater than 100, and less than 1000, it will also bo found in the column marked N, and the- decimal part of its logarithm opposite to it, iu the column at tlie top of which is 0. If the number contains 4 digits, the first three of them will be found in the column under N, and th« fourth at the top of the pngo ; and tlion its logarithm in Jic same horizontal lino as the thi(!o first digits of the given nunibor, and in the same column as its fourth ill v 322 LOGARITHMS. ^ If the number contains more than 4 digits, find the logarilhui of its first, four, and ;ilso the diifcrence be- tween that and the h)gaiithni of tiic next higher num- ber, in the table ; multiply this diiFereuue by the remain- ing digits, and cutting off from the pr(^uct so many digits as were in the multiplier (but at the same time ftdding unity if the highest cut off is not less than 5), add it to thcr logarithm corresponding to the four first digits. Example 1.— The logarithm of 59 is !• 770852 (the charac- teristic being positive, and 07w less than the number oiintegers) . Example 2.— The logarithm of 338 is 2528917. Example 3.— The logarithm of -0004587 is — 4(561529 (tlie characteristic being negative, and one moix than the number of prefixed cyphers) . Example 4.— The logarithm of 28434 is 4-453838. For, the difference between 453777 the logarithm of 2843, the four first digits of the given number, and 453930 the logarithm of 2844, the next number, is 153 ; which, multi- plied by 4, the remaining digit of the given number, pro- duces G12: then cutting off one digit from this (since we have multiplied by only one digit) it becomes Gl, which being added to 453777 (the logaritlim of 2844) makes 453838, and, with the characteristic, 4453838, the required logarithm. Example 5.— The logarithm of 873457 is 5-941242. For, the difference between the logarithms of 8734 and 8735 is 50, which, being multiplied by 57, the remaining digits of the given number, makes 2850; from this we cut off two digits to the right (since we have multiplied by two digits), when it becomes 28 ; but as the highest digit cut off is 5, we add unity, which makes 29. Then 5-941213 (the logarithm of 8734) -[-29=5-941242, is the required logarithm. 39. Except when the logarithms increase very ra- pidly — that is, at the commencement of the table — the differences may be taken from the right hand column (and opposite the three first digits of the given number) where the mean differences will be found. Instead of multiplying the mean difference by the remaining digits (the fifth, &c., to the right) of the given number, and cutting off so many places frc^ the product as are equal to the number of digits in the multiplier, tx) obtain the iir-)pur!luaal part — or what is to be added 5, 2u. LOGARITHMS. 323 to the logaritlim of tlie first four digits, we may tako the ^-oportioiuil part corrcspouding to each of the re- inaitiing digits from that part of the columu at the left hand side of the page, which is in tlio same horizontal division as tliat in which the first three digits of the givvon number have been found. K.VAMi'Li:.— What is the logarithm of 839785 ? The (decimal part of the) logarithm of 8B9700 is 924124. Opposite to 8, in the same horizontal division of the page, wo lind 42, or rather, (since it ia 80) 420, and opposite to 5, 2u. Monce the re(iuirod logarithm'is 9241244-420-f2G=«: Vi24570: uud, with the characteristic, 5-924570. 40. Tlic mctliod given for finding tlie proportional part — or what is to bo added to the next lower logarithm, in the table— iirisos from tlie diiferoiico of numbers being proportional to the ditference of their logarithms. Hence, using the last example, 100 : 86 : : 62 (92417*i, the logarithm of 839800—924124, the logarithm of 839700) : ""Vqq-. or the difference (the ynean difference mnj generally be used) X by the remaming digits of tlie given nuiMl)or — 100 (the division being performed by cut- ting off two digits to the riglit). It is evident that the number of (iigits to be cut off depends on the nuniber of digits in the multiplier. The logaritlim found is not exactly correct, be- cause numbers are not exactly proportional to the difforcncca of their logarithms. The proportional parts set down in the left hand column, have been calculated by making the necessary multiplica- tions and divisions. 41. To find the logaritjiini of a fraction — lluLE. — Find the logarithms of both numerator and denominator, and tlien subtract the former from tho latter ; this will give the logarithm of the quotient. Example.— Log.' i| is 1-672098 - 1-748187 = - 1-923910. Wo find that 2 is to be subtracted from 1 (the character- istic of the numerator) ; l)nt 2 from 1 leaves 1 still to bo snbtractcd, or [Sect. II. 15 j — 1, the characteristic of tho quotient. Wo shall find presently tliat to divide one quantity bj anoth.or, avo have merely to subtract the lop;arithm of the I'attei iVoni that of tho former. 42. To find the logarithm of a mixed number — iluT.K. — llcduee it to an improper fraction, and pro c<"od as directed by the last rule. |i| /'■ 324 LOOARITIIMS. 43. To fiud the numjicr which corresponds to a given logarithm — • '"• If the logarithm itself is found in the tahle — lluLE. — Take from the table the number which cor- responds to it, and place the decimal point so that there may be the requisite number of integral, or decimal places — according to the characteristic [37]. Fa' AMPLE. — What number corresponds to the logarithm 4-214314? AVo find 21 opposite the natural number 103 ; and look- i -g along the horizontal line, we find the rest of the logarithm under the figure 8 nt the top of the page : therefore the digits of the required number are 1038. But as the charaeteribtic is 4. there must in it be 5 places of integers. Hence the required number is 1G380. 44. ,If the given logarithm is not found in the table — ■ liuLE. — Find that logarithm in the table which is next lower than the given one, and its digits will bo the highest digits of the required number ; find tho diflerence between this logarithm and the given one, annex to it a ryphcr, and then divide it by that differ- ence in the table, which corresponds to the four highest digits of the required number — the quotient will be the next digit ; add another cypher, divide again by the tabular diflerence, and the quotient will be th-: next digit. Continue this process as long as necessary. ]'2xAi\rPLE. — What number corresponds to the logarithm 5054329 1 C54273, which corresponds with the natural number 4511, is the logarithm next less than the given one ; therefore the first four digits of the required number are 4511. Adding a cypher to 50, the difference between 054273 and the given logarithm, it becomes 500, which, being divided by 90, tlie kihidar difference corresponding with 4511, gives 5 as quo- tient, and 80 as remainder, I'herefore, the first five digits of the required number are 45115. Adding a cyplier to 80, it becomes 800; and, dividing this by 90, we obtain 8 as the next digit of the required number, and 32 as remainder, "^riio iiife'rcrs of the required numl^or (one more than 5, tho characteristic) are, tlierofore, 451158. We may obtain the decimals, by continuing the addition of cyphers to the re- mainders, and the division by 90. 4o. V X. LOQARITIIMa. 325 45. Wc arrive at tlio same lesult, by Bubtracting from .the difference between the given logarithm and the next less in the table, the highest (which doesno exceed it) of those proportional parts found at the right W side^f the page and in the same honzontaWm- sion with the first three digits of the given number-- continuing the process by the addition of cyphers, until nothing, or almost nothing, remains. FxAMPLK.-Usin- the last, 4511 is the natural number cor esTonUng to the logarithm G54273, which differs from he ^enlolarithmby'sG. The Pvop-t-f.r^^^^^ %' B-imo horizontal division as 4511, are 10, 19, 2J, c5», 4», oo, G7 77 md 80. The highest of those, contained in 56, is 48. w I'ich we find opposit^e to, and therefore corresponding with tho natural number 5; hence 5 is the next of the •0 ulred digits. 48 subtracted from 5G, leaves 8 ; this, when a'^^pher is^dded, becomes 80, which contains 77 ^corres- w>udino- to the natural number 8)5 therefore 8 is the next ^ the "required digits. 77 subtracted f-m ^O, kaves 3 tliiK when a cypher is added, becomes 30, &c. ^^^o inte to 5 Therefore, of the required number, are found to be 451158, the sauie as those obtained by the other method. The rules for finding the numbers corresponding to civon loo-arithms are merely the converse of those used for finding the logarithms of given numbers. Use of Logarithms in Arithmtic. 46. To multiply numbers, by means of their loga- " Kml-Add the logarithms of the factors ; and the natural number corresponding to the result will be the required product. ExAMPLF..-87x24=1939519 (the log. of 87) -f 1-380211 Ohe^olof 24)=3319730; which i^fo^^^T ° ^rXx ivith the natural number, 2088. Therefore 87x24=2(588. from the very nature of indices. Thus f X° — ^0-'^,?^'^O^'^ jrom uie vLi^y and the abbreviation for multii.liodSXoXoXoXoXoxoxJ, -I indices (logatulnlv "The rule rnighl in the same way, be proved correct by any other example. p Q I 326 L0CiAlUTJIM8 47. When tho clmractcristies of tlio logarithms^ to be added arc both i)ositivo, it is cvidont that their sum will bo positive. When thoy are both negative, their sum (diminished by wliat is to bo carried from tlie sum of th.i positive [36] decimal parts) will be negative. When one is negative, and tho other positive, subtract tho less from the greater, and prefix to the difference the Bign belonging to tho greater — bearing in mind what has been already said [Sec. II. 15] with reference to the subtraction of a greater from a less quantity. 48. To divide numbers, by means of their logarithms — liui.R. — Subtract tho logarithm of the divisor from that of the dividend ; and tho natural number, corres- ponding to tho result, will be tho required quotient. Example.— 1134 -f. 42=3054013 (the 1o;t. of 1184) — 1G23249 (the log. of 42) = 1-431304, which is found to correspond with the natural number, 27. Therefore 1134— 42=27. Reasox of the Rule. — This mode of division arises from the nature of indices. Thus 4*-i-4'=[2] 4x4X4X4X4— IX 4X4X4X4X4 ^ ^ 4x4x4 ^ 4X4= 4>^4x4 — =4X4X4^-^1=4x4, the abbreviation for which is 4»; Bu*, 2 is equal to tho index (logarithm) of the dividend minus hat of the divisor. Tlie rule might, in the same way, be provvd correct by any other example. 49. In subtracting tho logarithm of the divisor, if it is negative, change the sign of its characteristic or inte- gral part, and then proceed as if this were to be added to the characteristic of tho dividend ; but before making the characteristic of the divisor positive, subtract what was borrowed^ (if any thing), in subtracting its decimal part.^ For, since the decimal part of a logarithm is positive, what is borrowed^ in order to make it possible to subtvact the decimal part of the logarithm of the divisor from that of the dividend, must be so much taken away from what is positive, or added to what '3 negative in the remainder. We chac^ge the sipin of tho negative characteristic, ana then add it; for, adding a positive, is the same as taking awny a negative quantity. V Ill i,o«AurrHM«. fsrr in ^0. To -also a quantity to any power, by means of it»- logaritli.n — lluLE. — Multiply tho lofrarltlim of tlie qiianity by tho index Oi t\w power; and the natural number cor- rcbponding to the re.sult will be the required power. KxAMPi.E. — Puufic 5 to tho 5th power. The lo^'arithm of 5 is OO'.IHOT, whieh, niultlpllcd by 5, gives 3'4'J486, the logarithm of 3125. Theroforo, the 5th jiowcr of 5' is 3125. kr.AHON OK Tiiii Rui.K. — Tliis vnlc also follows from tlie nawire of imlicea. o* vnisod to tlie otli power is 6X& inuUi])li»>d hy Hx'o )uulti{)lieil by oX5 nuillii;licil by SX-) nmltiplieil by rjX5, or r)X'''X5xr)X''iX'''>Xi)X'^X':>X0, the abbreviation for which is [2] 5'". lUit 10 is eiiiial to 2, the index (b.garithm) of tho quantity, niultiplioa by 5, that of the ])Ower. The rnle might, in the same way, bo proved correct by uuy other example. 51. It follows from what has been said [47] tbitt wbcu a negative chariieteristic is to be multiplied, the produet in nrrrativc ; and that what i:< to be carried from the nmltijljcation of the decimal part (always positive) is to be suhtraclcd from tlii.s mj^ativo result. 52. To evolve any quantity, by means of its loga- rithm — Bulk. — Divide the logarithm of the given quantity by that number which expresses the root to be taken ; and the natural number corresponding to the result will be the re(|uired root. Ex.ypi.E.— What is the 4th root of 2401. Tlie lo;i;arithm of 2401 is 3 o80302, which, divided by 4, the number expressing the root, gives -845098, the logarithm of 7. Therefore, the fourth root of 2401 is 7, Heason ok the Rule. — This vule follows, likewise, from tho nature of indices. Thus the 5th root of ItV ia such a number as, raised to the 5th power— tliat is, taken 5 times aa <i factor— would produce 16'". But loV, taken 5 times as a factor, would produce 10'". The rnle might be prove 1 correct, equally well, by any other example. 53. "When a negative characteristic is to bo divided— UiTi.i: I.— If the cliaractevistic is cKadly divi.sible by the divisor, divide in the ordinary way, but make tho characteristic of the quotient negative. I III 1 A 1^4 m Wm 328 LOGARITHMS. TT. — Tf llio nogativo clmractcristio is nni exactly rlivisiblc, juJd wluit will iimko it so, both to it and to the decimal part of tho logarithiu. Theu proceed with tlu- division. Example.— Divide tho logarithm —4' 887564 by 5. 4 w antH 1 of bciug divi«iblo by 5; then — 4•8375C4-^5=a — 5-f.l-8375G4-i-5=13G7513, tho required logarithm. * Rraaon of I. — Tho quotient multiplied by tho divisor must give tlio dividend; but [61] a ncgativo quotient multiplied by a positive divisor will give a negative dividend. Kkahon of II. — In cxanii>le 2, avc luivo merely added -f- 1 and — 1 to tho same quantity — wliich, of course, docs nC- .alter it. QUESTIONS. 1 . What are logarithms ? [32", . 2. How do they facilitate calculation .? [33] . 3. Why ia a table of logarithms necessary } [34]. 4. What is the characteristic of a logarithm ; ant/ how is it found ? [37] . 5. IIoAv ih the logarithm of a number found#by tho table.? [38]. f). How are the " diflferenccs," given in the tablo used.? [30]. 7. What is the use of "proportional parts .?" [39]. S. How is the logarithm of a fraction found ? [41]. 9. How do we find the logaritlim of a mixed num- ber ? [42] . 10. How is tho number corresponding to a given logarithm found ? [43] . 11. How is a number found when its corresponding logarithm is not in the table ? [44]. J2. How are multiplication, division, involution and evolution effected, by means of logarithms r [46, 48, 50, and 52]. 13. When negative characteristics are added, what is tlie sign of their sum ? [47]. 14. What is tlio process for division, when tho cha- racieristic of the divisor is negative ? [49] . 15. How is ancgTitivc characteristic umltiplied r [51]. 10. How is a negative charactori.stio divided .? [53] 5 5 329 SECTION X. PROGRESSION, &o. 1. A profTreasion consi.sts of <*i nnmbnr of quantities Jill cm a sing, or decreasing l)y fi ciirtaiu law, and forming ^vliat uro called cmitmued propor/Umnts. When the terms of the series coui*tantly increase, it is said to l>«) an ascending^ but when they decrease (increase to (he /rft), a descending scries. 2. in an fqnidiO'crcnt or a ?-i/Awt'/(!"mZ progression, tho qnantities inci-ease, or decrease by a annvion difference. Tiius 5, 7, 9, 11, &.O., is an ascending, and 15, 12, 9, 6, &c., ia a descending arithmdical series or progrtission. The common diflerencc in the former is 2, and ni the latter '^. A continued proportion may bo formed out T)f such a series. Thus — 5 : 7 : : 7 : 9 : : 9 : 11, &c. ; and 15 : 12 : : 12 : 9 : : 9 : 0, &o. Or we may say 5 : 7 : : 9 : 11 : : &c. ) and 15 : 12 :: 9 : 6 :: &c. 3. In a geometrical or equirallonal progression, tho quantities increase by a common ratio or multiplier. Thus 5, 10, 20, 40, &c. ; -and 10000, 1000, 100, 10, &c., are geometrical series. The common ratio in the former case is 2, and the quantities increase to the right ; in the latter it is 10, and tho quantities increase to tho left. A continued proportion may be formed out of Huch a series. Thus — 5 : 10 : : 10 : 20 : : 20 : 40, &c. ; and 10000 : 1000 : : 1000 : 100 : : 100 : 10, &c. Or wo may say 5 : 10 : : 20 : 40 : : &c. ; and 10000 : 1000 : : 100 : 10 : : &o. 4. The first and last terras of a progression are called its extremes, and all the intermediate terms its means. 5. AritliMeiiad rrogression.—To find the sum of a series of terms in arithmetical progression — j^^yij,;, — Multiply the sum of the extremes uj nan the number of terms. I 330 PROGRESSION. _ Example.— Whafe is the sum of a series of 10 terms, tlio tirat being 2, and last 20 '? Ans. 2-f 20x •2-'=110. C. Reasoiv of the RuLic.— This rule can be easily proved. For tins purpose, set down the progression twice over— but in sucli a way as that the last term of one shall be under the nrst term of the other series. Then, 24+21+184-154-12-f 9=the sum. 9-f 12-f-15 4-18-i-21+24=the aum. And, adding the equals, 3;J+33-j-334-33+33+33=twice the sum. That is, tf^ir.e tlie sum of the series will be equal to the sum of aa ii.any quantities as tliere are terms in the series— each of the quantities being equal to tlie sura of the extremes. And the sum of the series itself will be equal to half as much, or to the sum of the extremes taken ha/f as many times as tliere are terms in flie series. The rule might be proved correct by any other example, and, therefore, is general. EXERCISES. 1. One extreme is 3, the other 1.5, and the number of terius is 7. What is the sum of the series ? Ans. 63. 2. One extreme is 5, the other 93, and the number of terms is 41). ^VHiat is the sum .? Ans. 2401. 3. One extreme is 147, the other |, and the number* of terms is 97. What is tlie sum ? Ans. 7165-875. 4. One extreme is 4^-, the other 143, and the num ber of terms is 42. What is the sum > Ans. 3094-875 7. Given the extremes, and number of terms — to fini the common difference — lluLE. — Find the difference between the given ex- tremes, and divide it by one less than the number of terms. The quotient will bo the common difference. Example.— In an arithmetical series, the extremes are 21 and o, and the nunibcr of terms is 7. Wha^; is the common diilerence '? 21 — 3-^7 — 1 = 18-i-6 = 3, the required number. 8. Reasov of thk Rule.— The diflference between the greater and lesser extretne arises from the common ditferenco. bemg aiJded to the lesser extreme once for every term, ex- cept tbe lowest ; that is, the greater contains the lesser extreme plus the common difference taken once less than the number of terms. Therefore, if we subrract the lesser from the greater extrcriiu, the diifereuce oblaiuod will be equal to the common dil}vM-en'»e multiplied by one less than the number of terms And if wo divide tlio difference by one less than the number of tcrm.^ we will have the cuininun difference. 5. Tb and 497, common 6. Th and 9|, : common 7. Th and I, a: common 9. To two give Rule cording it to, or term ; a the thirc ing tern Wen terms is EXAMI 21. 21- the seri( 6 . Exam 10. 30 the Borl 8. r A:n3. 4 9. 1 Ans. 1 10 Am. 6 PnOftRESSION. 331 EXERCISES. 5. The extremes of an arithmeti(;al series are 21 and 497, and the number of terras is 41. What is the common difference .? Aiis. 11-9. ^ m-ro, 6 The extremes of an arithmetical serieR are 127«a and 91, and the. number of terms is 26. What is the common difrerenee ? Aiis. 4^. ^ 7 The extremes of an arithmetical series are 77|f and*!, and the number of terms is 84. What is the common difference ? A^is. |f . 9. To find a7ii/ number of arithmetical means between two given numbers — Rule.— Find the common difference [7] ; and, ac- cordinc' as it is an ascending or a descending series, add it to, or subtract it from the first, to form the f oond term ; add it to, or subtract it from the second, to torni the third. Proceed in the same way with the remain- ing terms. , , i. <? VVe must remember that one less than the number ot terms is one more than the number of means. Example l.~Find 4 arithmetical means between 6 and 21. 21-6 = 15. TTT==''^) th^' common difference. And the series is — „ « . r o 6.6+3. 6+2x3 . 6+3x3 . 6+4x3 . 6+5x3. Or 6 . 9 . 12 . 15 . 18 . 21. Example 2.— Find 4 arithmetical means between 30 and 10 30-10=20. TITT^^' *^^ common difference. And 4+i the Bcrics ia — io ^ i i in 30 . 26 . 22 . 18 . 14 . 10 This rule is eyideut. EXERCISES. 8 Find 11 arithmetical means between 2 and 26 A:m. 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and 24. 9 Find 7 arithmetical means Detween 8 and 64 Ans. 11, 14, 17,20, 23,26, 29. 10 Find b arithractieal means between 4J-, and id^ Am. 6,7J,9, 101, 12. 332 rUOGRESSION. 10. Given the extremes, and tlie number of terms — to find any term of an arithmetical progression — Rule. — Find the common difference by the last rule, and if it is an ascending series, the required term will be the lessor extreme plus — if a descending series, the greater extreme minus the common difference multiplied by one less than the number of the term. Example 1. — What is the 5th term of a series containing 9 terms, the first being 4, and the last 28 ? 28-4 ^ — Q — =3, is the common difference. And 4-f 3x5 — 1= 16, is the required term. Example 2. — What is the 7th term of a series of 10 terms, the extremes being 20 and 2 ? 20-2 ^ _ „ — n— =2, is the common difference. 20 — 2x7—1=8. is the required term. 11. Reason OF the Rxtle. — In an ascending series th» required term is greater than the given lesser extreme to the amount of all the differences found in it. But the number of differences it contains is equal only to the number of lerir.a which precede it — since the common difference is not found in the^trs^ term. In a descending series the required term is less than the given greater extreme, to the amount of the differences sub- tracted from the greater extreme — but one has been subtracted from it, for each of the terms which precede the required term. exerclses. 11. In an arithmetical progression the extremes are 14 and 86. and the number of terms is 19. What is the 11th term .? Ans. 54. 12. In an arithmetical series the extremes are 23 and 4j and the number of terms is 7. What is the 4th ttrm } Ans. 13. 13. In an arithmetical series 49 and £ are the ex- tremes, and 106 is the number of terms. What is the 94th term } A^is. 6-2643. 12. Given the extremes, and common difference— to find the number of terms — 11 ule.— 'Divide i\\Q differcnco bctAvccn the given ex- tremes by the common difference, and the quotient plus unity will be the number of terms. Ex A Mr 1 which th ence 3 '? 20- 3 13. He, lesser ext terms, i except th the extre ■will be C3 14. Ii and 12, number 15. Ii and 32, number 16. I 13 '<.}•) an I number 14. G and one llULE terms, n The difi EXAMI the nun AVhat is 15. Rj Slim = 8 divide ec vre shall 2 X the num tractiug have rKOGREHSION. 833 ExAMiM.K. — How many tenns in jin firUhinotical series of which tho extromea arc 5 and 20, and Iho common ditfer eiicc 3 ? 20-5 3 =7. And 74-1=8, is tlie number of terms. 13. llEAscN OF THE Rui.E.— The srcatcr diifcrs from tho lesser extreme to tlie amount of the diircrcncos found in all the terms. But th» common difference is found in all the terms except the lesser extreme. Therefore the difference between the extremes contains the common difference once less than will be expressed by the number of terms. EXERCISES. 14. In an arithmetical series, tlie extremes arc 96 and 12, and tlie common difference is 6. What is tha number of terms ? Ans. 15. 15. In an arithmetical series, the extremes are 14 and 32, and the common difference is 3. What is the number of terms } Ans. 7. 16. In an arithmetical series, the common difference is A, and the extremes are 14f and il. What is the number of terms ? Ans. 8. 14. Given the sum of the series, the number of terms, and one extreme — to find the other — lIui.E. — Divide twice the sum by the number of terms, and take the given extreme from the quotient The difference will be the required extreme. Example.— One extreme of an arithmetical scries is 10 the number of terms is 6, and the sum of the series is 42 What is the other extreme 1 2X42 -10 = 4, is the required extreme. 15. Reason of the Rule.— We have seen [5] that 2 X the sum = sum of the extremes X the number of terms. But if wo divide each of these equal quantities by the number of terms, ■we shall have , o. 2 X the sum sura of extremes X the number of terms the number of terras 2 X the sum the number of terms , — „ =s sum of the extreme!?. .And sub- the number oi terms tractiug the same extreme from each of these equals, wo shall have ur H34 i>K()(;kxs«u)!v. „ X ''"^J^"J^ — onocxtioaie=the sum of the extremes the number of tonus tlie aaiue extreme. twice the sum Or ;^i.: „i rrr.:F7;-~:z minus one extreme = the other ox- tho number of terms a-erae. EXERCISES. 17. One extreme is 4, the number of terms is 17, /md the sum of the series is 884. What is the other extrcuio ? Ans. 100. 18. One extreme is 3, the number of terms is 63, and the sum of the series is 252. What is the other extreme ? A71S. 5. 19. One extreme is 27, the number of terms is 26, and tlie sum of the scries is 1924. What is the other extreme.? Ans. 121. 16.^ Geometrical Progression. — Given the extremes and common ratio — to find the sum of the series — Rule. — Subtract the lesser extreme from the product of the greater and the common ratio ; and divide the difference by one kss than the common ratio. Example. — In a geometrical progression, 4 and 312 are the extremes, and 'he common ratio is 2. What is the sum of the series. 312x2 — 4 — 2 — Y — ~ ^= ^^^' *^® required number. 17. Reason of the Rui^e. — The rule may be proved by setting down the series, and placing over it (but in a reverse order) the product of each of the terms and the common ratio. Then Sum X common ratfo = 8 4- 16 + 32, &c. . + 81 2 -f- 624 Sum= 4 + 8 + 16+32, &c. . 4-312 . And, subtracting the lower from the upper lino, we shall have Sum X common ratio — Sum = 624 — 4. Or Common ratio — 1 X Sum = 624 — 4. And, dividing each of the equal quantities by the common ratio minus 1 642 (last term X comrrfon ratio") — 4 (the firgt term) Sun = ^^- ^ vv— — 5, — ^^ ~ common ratio — 1 Which ib the rule. 20. The 2, and th Ans. 682. 21. The 175092, an Ans. 1932 22. The ore yV anc the sum.? Since the 23. The 937-5, and A.)is. 1171 18. Giv goometricii RULE.- by thc! les! i.s indicate bo the req EXAMPI-F pro<:;rcs8U)v 11 1011 ratio 'i SO 19. Tiv.^u to the Icsse tlie comuioi' pinuo the c> is, tlie gren power imlic tiplio'l by t by the I0S.S1 is indicate; obtain the 24. Th common 1 85. Til I'R0(iRE.S3I0N 335 EXERCISKS. 20. The extremes of a geometrical pciies are 512 and 2, and the common ratio is 4. What is the sum ? Ans. 682. 21. The extremes of a geometrical series are 12 and 175092, and the common ratio is 1 1. What Is the sum ? Ana. 193200. 22. The extremes of an infinite geometrical series ore yV and 0, and j\ is tlie common ratio. What is the sum.? Ans. -i. [Sec. IV. 74.] yinco the sorics is infinite, the lessor extreme=0. 23. The extren\es of a geometrical series are *3 and 937-5, and the coiumou ratio is 5. What is the sum ? A)is. 1171-875. 18. Given the extremes, and number of terms in a geometrical series — to find the common ratio — EuLE. — Divide the greater of the given extremes by the lesser ; and take that root of the quotient which i.s indicated by the number of terms minuni 1. This will be the required number. ExAMPi-E.— 5 and 80 arc the extremes of a geometrieal progression, in which there arc 5 terms. ^Vhat i^ the eoiu- 11 ion ratio ? 7-=lG. And ^'1G=2, the required common ratio. o 19. REASON OT THK Rule. — The greater extvemc is o.i|nnl to the lesser multiplied by a product whicli has for its factors tlie conmioii ratio tukeri once loss tiian the number of terms — since the comni<in ratio is not found in tiie Jiist term. That is, tlie trreater cxtre\ae contains the common ratio raised to a power indicated by 1 less than the number of terms, and mul- tiplied by the lessor extreme. Consequently if, after dividinj; by the lessor extreme, we take that root of the quotient, which is indicated by one less tlian the number of terms, we sholl obtain the common ratio itself. EXERCISES 24. The extremes of a geometrical series are 4911^2 oj^fl 3^ atul the number of terms is 8. What is th3 common ratio } Ans. 4. 2d. The extreuies of a geometrical series arc 1 and 336 PROGRKSSION, 15625, and the number of terms is 7 What is the common ratio ? Ans. 5. 26. The extremes of a geometrical series arc 20176S035 and 5, and the number of term* is 10 What is the common ratio ? Ans. 7. 20. To find any number of geometrical means be two on two quantities — Rule. — Find the common ratio (by the last rule)^ and — accovdina; as the series is ascendinoj, or descend- ing — multiply or divide it into the first term to obtain the second ; multiply or divide it into the second ta obtain the third ; and so on with the remaining terms. We must remember that one less than the number of terms is one more than the number of means. Example 1. — Find 3 geometrical means between 1 and 81., ^--=:3, the common ratio. And 3, 9, 27, are the re- quired means. E-XAiMPLE 2. — Find 3 geometrical means between 12r)f und 2. 1250 _ . . 1250 1250 1250 4/-2-=''>- And H, or 250, 50, ]';' 5 5x5 5x5x5 are the required means. This rule requires no explanation. EXERCISES. 27. Find 7 geometrical means between Sand 19683 ' Ans. 9, 27, 81, 243, 729, 2187, 6561. 28. Find S geometrical means between 4096 and Si" Ans. 2048, 1024, 512, 256, 128, 64, 32, and 16. 29. Find 7 geometrical means between 14 and 23514624.? Ans. 84, 504, 3024, 18144, 108864, 653184, and 3919104. 21 . Given the first and last term, and the number of terms — to find any term of a geometrical series — lluLE. — If it bo an ascendhig series, multiply, if a df'sp.or>diTi'» sorins). divJ;<ft tho firs.t, tomi hv> fl 'ij imwrrr of the common ratio which is indicated by t;*; num&ei of the term minus 1. is the 30. 32. PROGKESSION. 337 Example 1.— Find tlio 3nl term of a {reometrical scries, of %vhich the tiryt term is (3, the last 1458, suid the number ol" term.-) G. 1458 The common ratio is ;j/-^-=P). Tlicveforc the required term is 0x3^=54. ]'>xAMPi,K 2. — Find the 5th term of a series, of which tlw extremes are 524288 and 2, and the nundjer of terms is 10. 5242SH . 5242NS Iho common ratio ^ — rr~— 4. And — j4- = 2048 is the required term. 22. Rkasoiv of the Riti.k.— Tn an ascending series, any term is the proihict of the lirsi and the couinion ratio taken as a factor so many times as there arc preceding terms — siiKsa it is not found i7i ihe fust term. In ade-ccndiug series, nny term is eqnal to the first term, divided by a product containing the common ratio as a factor so many times as tliero arc pi-eeoding terms — since evei-y term but that whieh is required adds it once to the factors wducli coDstitute the divisor. EXERCISES. 30. Wh;it is tho 6th term of a series having 3 and 5851)375 as extremes, and, containing 10 terms .^ Ans. 9375. 31. Given 39366 and 2 as tho extremes of a series Laving 10 terms. What is the 8th term.? Ans. 18. 32. Given 1959552 and 7 as tlio extremes of a series havinff 8 terms. What is the Gth term ? Ans. 252. O 23. Given the extremes and common ratio — to find the number of terms — Rule. — Divide the greater by the lesser extreme, and one more than the number expressing what power of common ratio is equal to the quotient, will be tho re(|ulrcd qiumtity. ExAMi'Lic. — How many terms in a series of which tho extriMnes are 2 and 25G, and the conuuou ratio is 2 ? -,-^-=128. But 2"=::128. There are, therefore, 8 term£3. The common ratio is fonnd as a factor (in the quotient of tho greater divided by tho lo.'^ser extreme) once less than the number of terms. II L 838 PROGRESSION. KXERCISES. 33. TIow many terms in a xories of wliich the first, is 78732 and the last 12, and the cummun ratio is 9 .-* Ann. 5. 3-4. IIoAV niany terms in a series of wliieli llic ex- tremes and common ratio are 4, 47()r)9f;, and 7 ? Avs. 7. 35. How inany terms in a series of Avliieh the ex- tremes and common ratio, are 19GGUb!, G, and 8 f Ans. 6. 24. Oiv^en the common ratio, number of terms, and one extreme — to find the other — lluLE. — If the lesser extreme is given, multiply, if the greater, divide it by the common ratio raised to a power indicated by one less than the number of terms. ExajMpi.k 1. — In a g;oomctrioal series, the lesser extreme is 8, the number of terms is 5. and the enniinoii ratio is Gj what is the other extreme '? Atis. 8xG'~'=10oG8. Example 2. — In a goomefrioal series, the greater extreme is 0561, the number of terms is 7, and the common ratio is S; what is the other extreme'? Jnx. G5Gl-j-3'~'=U. This rule does not require any exphvnatiou. EXERCISES. 36. The common ration is 3, tlio number of terms is 7, and one extreme is 9 ; what is the other ? Ans. 0561. 37. The common ratio is 4- the number of terms is 6, and one extreme is 1000 ; what is the other ? Ans. 1024000. 38. The common ratio is 8, the number of terms i** 10, and one extreme is 402653184 ; what is the other ? Ans. 3. In progression, as in many othe- rules, the application of algebra to the reasoning '.voukl greatly simplify it. MISCELLANEOUS EXERCISES IN PROGRESSION. 1. The clocks in Venice, and some other places stiilvO the 24 hours, not beginning again, as ours do, after 12. How many strokes do tliey give in a day ? Ans. 300. 2 A butcher bought J 00 sheep; for the first ho gav(' 16". , and for the larit i.'9 lO.:;. What did ho pny for Of Wl all, suppi Ans. £d 3. A yard he price of ; 4. A] the first on, until did he tr 5. A that the and that year. I 6. Fii Ans. ' 7. 8. payment being £ common the ratio 9. Wl thing fo: second, shoe .'' 10. A queathec gave £1 next, li was the of the I ceived i 1. W series ? 2. W trical pi names .^ 3. W ratio ? I PROGRESSION. 339 nil, supposiig tlicir prices to form an arithmetical scries ? Ans. iioOO. 8. A person bought 17 yards of cloth ; for the first yard he gave 2.?., and for the last IOj. What was the price of all ? Ans. £r> 2s. 4. A person travelling into the country went 3 miles the first day, 8 miles the second, 13 the third, and so on, until ho went 58 miles in one day. How many daya did he travel? Aois. 12. 5. A man being asked how many sons he had, said that the youngest was 4 years old, and the eldest 32, and that he had added one to his ftimily every fourth year. How many had he ? Ans. 8. 6. Find the sum of an infinite series, -J-, ^j -gVj &c. Ans. 1. 7. Of what value is the decimal -463' ? Ans. ^f f . 8. What debt can be discharged in a year by montlily payments in geometrical progression, the first term })eing jei, and the last £2048; and what will be tho common ratio ? Ans. The debt will be £4095 ; and the ratio 2. 9. What will be the price of a horse sold for 1 far- thing for the first nail in his shoes, 2 farthings for the second, 4 for the third, &c., allowing 8 nails in each shoe ? Ans. £4473924 55. 3frZ. 10. A nobleman dying left 11 sons, to whom he be- queathed his property m follows ; to the youngest he gave £1024; to the next, as much and a half; to tho iiext, 11 of the preceding son's share ; and so on. What was the eldest sou's fortune ; and what was the amount of the nobleman's property r Ans. The eldest son re- ceived £59049, and the father was worth £175099. QUESTIONS. 1. Wliat is meant by ascending and descending series ? [1]. 2. What is meant by an arithmetical and geome- trical progression ; and are they designated by any other names ? [2 and 3] . 3. What are the common difierence and common ratio } [2 and 3] . i II 540 ANNf ITIKS 4 t\).\>^ that a onntiimofl proportion mny 1)0 fomied froTi, a 8crics of ehhov kiiul r [2 iind 3J. .'j. Wliat arc moans ii"/i (^xtrcniej ? [4]. 6', l^ow is the hm, aritniuetical or a gcomo* trical series fouiivl ? jo auu Itij. 7. [low is the common difference or common ratio found ? [ 7 and 1 8] . 8. How is any nimiber of aritliraetioal or geometrical means fouwd ? [9 and 20] . 9. How is «ny particular arithmclicul or geometrical mean found ? [10 and 21]. 10. How is tlio number of terms in an arithmetical or geoinctriual series found .? [12 and 23]. 11. How is one i^xtremo of an arithnuitical or geome- trical scries found } [14 and 24]. ANNUITIES. 25. An annuity is an income to he paid at stated times, yearly, half-ye;irly, &o. It is either in possession ^ that is, entered upon alread}'', or to he entered upon immediately ; or it is in reversion^ that is, not to com- mence until after some period, or after something has occurred. An annuity is certain when its commence- ment and termiuation are assigned to definite periods, conlingcMt when its l)Oginning, or end, or both are uncertain ; is in arrears when one, or more payments are retained after they have become due. The amount of an annuity is the t.im of the payments forborne (in arrears), and the interest duo upon them. When an annuity is paid off at once, the price given for it is called its ]n-escnl: worthy or value — which ought to be such as would- if left at compound interest until the annuity ceases — 'produce a sum fqual to what would be due from the annuity left unpaid until that time. This value is said to l;»^e so many years'* purchase ; that is, so many annual payments of the income as would "be just equivalent to it. 26. To find the amount of a certain number of pay- ments in arrears, and the interest due on them — iJlJT.F..— the sum of bo the vci[\ EXAMPI.K uuiialdfor C The Inst, them, form 4 . . XlX^^ Example is unpaid 1 por emit, p In this c with its in luulli plied with its ill Ik the %\\i ''i'lie unu tlio trouti^■ the series X2-100U2) • ~1'( The am same aa t years, whi sov, is eq Xl; that the roquii It vou ar.nuiiV ! 2-527 - to bo CO 1 Hence payment them — Subtr number interest tirnt by ANNUITIES liUT.F.. 341 llion aud interest duo on tlicm, will -Find the interest dtie on each payment payinoi anioun uin, Iho smii of the p Ic tho reiiUKtrd Faampi.k 1.— Wliat will he the amount of XI .per ann ui mafd for G years, 5 per cent, simple interest heing allms .dT The last, and , ^-oeedingpaymeni.^vith the inteij^s^^ theui, form tho .'. ^tfnnetical sov.e. ^'-^.^''^^^^^-^ 4 XlxXOS XI. Audits sura 18 X14-X1+-^^'>X'X |=X24-X'25X3=XG'75=X0 15.s., the required amount. ExAMi'LK 2.-If the rent of a {\irm worth XGO r;;^ a.nnuni is unpaid fo^ 10 year., how much doen u amount to, at . pt>r cent, per an. compound mtercst . In thi« ease the series is g.o,..fr/ra/ ; and the last my^^en wilh its interest is the onunmt of XI for IH (IJ — i) J**''^" mvlli hod lY tho siven annuity, the pi;eeoding paynie with itH iutorvst is Iheamomit of Xi fnr 17 years mulUphed hy the ^^iven annuiiy, &:c. ^^^^ ^^^^^ ^,f t.^';vc:rTf^l^yiar:^:e2.^^^^^ Then tl. sum of the series is — ££i2!^^X^:2i^55ll'^l?[l()]=:l •i2-4, the required amount. The iMivmnt of XI f.a- 18 years mvdtii.lied hy 105 is the same as the Huu.imt ol XI toi U, oi ilo ol'' , the divi- vo'v^ which is found to he X252/ . And lOo —- 1, trie uuj ^;^is e ual to the amount of Cl for oue payment imnua 11'; thltis,tothe Intcre t of XL fc.; one payment. Hence X2-52( X <^3i^' _ £18o24. the required sum will he - " .q^ It would evidently ho th. same ^hin^ to consider^ the ar.nui;y as XI, and then multiply the re;-. It hy 00. lluu '■^l-Xni X G0 = X18"'2-4. For an annuity of XGO ought to ho GO times as productive as one of only XL • Hence, briefly, 1 find tlie amount of any nun.her of ptiynients in arrears, and the compoum intore.t dae on ^^'sl^aet £1 from the amount of ^1 for the giverj T,mph<u- of pavments, and divide the diilcrenco^by 4Jiq interest of £1 for one payment ; then um.lU^^)iy ine 'i">^.^ tient l>y the "fiyeii nni. I ^1 t. w ' ■ ANN urn KS. re?J*«?ij''.?H'' "*' 'V.1^I*^^^^«— E'icl» payment, with its Into- mn ; K I ^ co„s unto u H.juirnt, a.uu.mt ; „u I tlu, Hu.n dtte •» u<t bo the sum of tlie.vo aruounlM-wliiel, Vunu a ,/.«r Wn^ •ones, because of tlio deoreash.g interest, ari" ng fro m thf .lecreiw.ug number of times of pa^monl, ^ ® which 1 nfn « L " °^ ? 7'Mm./,ra/ series, one extreme of Which IS tiiehrst payment plus tlio interest due upon it at th« aitterence the interest on one payment due at the next. Hut when cowpounil interoat is allowed, what is due will hn the sum <.f a geoniHrical series, one extreme of wEx^sth« hrst payment plus the interest duo on it at tho a^t the otlor Jnr/hf •*f^"'?i'/"^* it« common ratio £ pi i'ts tter hI inteieat due on the hrat payment at the time of the last will bo the intoros due for one less than tho numLr of payments payment''"' '' "'' '"^ "° "'° '^«' "^^^^ '''' '^^'^ KoS EXERCISES. fn.^i^''''^' '" *^, '^^^'^'^"^^ «f ^'37 per unnu.u unpaid Ls ^llTios ^'' """*• ^'' """• '''"^^' ^°^"^'^'^- 2. What is tho amount of m annuity of i^loo to continue 5 yeai;s at 6 per cent, per an. conipouua inte- rest .'^ Ans.£bmUs.2\d. i- i i^ 3. What is the amount of an annuity of ^£350 to 4 What is the amount of ,£49 per annum unpaid Ans ^ilu 5 ^^^ ''x''^' ^^^'P^'^"^ ^"*^^est b^^i"g 'lUovved .? 28. To find the present value of an annuity— Rule.— Find (by the last rule) the amount of tho given annuity if not paid up to the time it will cease. 1 hen ascertain how often this sum contains the amount ot J^l up to the same time, at the interest allowed. Example.— What is the present worth of an annuity of S^^^uf^'itm: 18 ye,.s would amount t, ANNIHTCKH. 313 But. .CI put to intorost for 18 years at the same ruto would fimiaiut to X2-40GG12. Tliorolm-o — *''40(j(?>~" ~ ^® required value. The sura to be puiil for the annuity should evidently bo euch ns would produce the aiimo as the annuity itself, ia the same tima. liXEKCISES. f). What is the present worth of an annuity of £27, to be paid for 13 years, o per cent. coinpoTind interest being allowed ? Ans. £2j3 \2s. (j\d. 6. What is the present worth of an annuity of J6324, to bo paid for 12 years, 5 per cent, compound interest beiug allowed? Aiis. £2671 I3s. 10}f/. 7. What is the present worth of an annuity of .£22, to be paid for 21 years, 4 per cent, compound interest being allowed } Ans. £308 12s. lOd, 29. To lind the present value, when the annuity is in perpetuity — lluLK. — l)ivide the interest which £1 would produce in perpetuity into £1, and the quotient will be tno sum reijuirud to produce an annuity of £1 per annum in perpetuity. iMultiply the (|uutioiit by the number of pounds in the given annuity, and the product AtII be the required present worth. Example. — Wliat is the value of an income of £17 for ever * Let us suppose that XlOO would produce £b per cent, per an. for ever: — thou £i would produce £-05. Therefore, to produce £1, we roquire as many pounds as will be equal to the number of times £-05 i,<* contained in £1. But-7r?=ai £20, therefore £20 Avould produce an annuity of £1 for ever. And 17 times as much, or £20x17=340, which would produce an annuity of £17 for over, is the required present value. EXERCISES. 8. A small estate brings £25 per annum ; wh&t is its present worth, allowing 4 per cent, per .annum irste- rest .? Ans. £62.5. 9. What is the present worth of an incorae of £347 ; 1 I 344 ANNUITIES. in perpetuity, allowing G per cent, interest? Ans £5783 6s. S(L 10. What is the value of a perpetual annuity of £46, aUowiug 5 per cent, interest ? Arts. ^£920. _ 30. To find the present value of an annuity in rever- sion — lluLE. — Find the amount of the annuity as if it were forborne^ until it should cease. Then fi^d-what sum, put to interest now, would at that time produce the same amount. Example.— What is the value of an annuity of £10 per annum, to continue for 6, but not to commenr- for 12 years, o per cent, compound interest being allowed ? ^'^1 ""J^^ofe^ ^^^ ^'^^ ^ ye^^'« if left unpaid, would bo l'?'!^-u^-a'^]^} ' ^^'1 ^1 ^0"!^' i» 1^ years, be worth i^ii'ObUoO. Ihercforo M8-0VJI ll~08959~^"^ ^^' '^'■^•■> i^ ^^^^ required present worth. EXERCISES 11. what is the present worth of .£75 per annum, which is not to commence for 10 y^ars, but will con- tinue 7 years after, at 6 per cent, compound interest ? Am. £1.55 9.?. 7-^d. 12. The reversion of an annuity of £175 per annum, to continue 11 years, and commence 9 years hence, is to be sold ; what is its present worth, allowing 6 per cent, per annu.n compound interest ? Ans. £430 7*. }d. 13. What is the p^-esent worth of a rent of £45 per annum, to commence in 8, and last for 12 years, 6 per cent, compound interest, payable half-yearly, being allowed.' ^?w. £117 25. S^^. 31 When the annuity is contingent, its value depend.^ on the probability of the contingent circumstance, or circumstances. A life annuity is equal to its amount multiplied by ' the value of an annuity of £1 (found by tables) for tho given age. ^ The tables used for the purpose are calcu- l:i ted on principles derived from the doctrine of chances, observations on the duration of life in different circum- stances, the rates of compound interest, &c. POSITION. 345 QUESTIONS. 1 . What is fin annuity ? [25'j . 2. What is au annuity in possession — in reversion — certain — contingent — or in arrears ? [25]. 3. What is meant hy the present worth of an an- nuity ? [25] . 4. Plow is the amount of any number of payments in arrears found, the interest allowed being simple or ■compound ? [26] . 5. How is the present value of an annuity in posses- sion foimd ? [28]. 6. How is the present value of an annuity in per- petuity found ? [29] . 7. IIow is the present value of an annuity in rever- sion found .' [30]. ^' i 1 POSITION. 32. Position, called also the " rule of false," is a rule which, by the use of one or more assumed, but false numbers, enables us to find the true one. By means of it we can obtain the answers to certain questions, which we could not resolve by the ordinary direct rules. When the results arc really proportional to the sup- pof'ition — as, for instance, when the number sought is to be muUijflied or divided by some pi'oposed number ; or is to bo increased or diminished by itseJf, or by some given mnUijplc. or j)art of itself — and when the ({uestion contains only one p)-opositio7ij we use what is called single position, assuming only one number ; and tho (juantity found is exactly that which is required. Other- wise — as, for instance, when the number sought is to bo increased or diminished by some absolute number, which is not a knovrn multiple, or part of it — or when two propositions, neither of which can be banished, are con- tained iu the problem, we use douhk position, assuming itjuo numbers. If the number souiclit is, durins; tho process indicated by the question, to be involved or evolved, we obtain only au approximation to the quan- tity required. I Ill 346 POSITION. .'«. ^^ingk PosiUon.-~B.vhK. Assume a number, and perform with it the operations described in the question ; then say, as the result obtained is to the number nse(l\ so IS the true or given result to the number required. ExAMPLK.— What number is that whicli, being multiplied ^J -». '^y i, and by 9, the sum of the results shall be 231 ? Let us assume 4 as the quantity sought. 4x5-j-4x74- 4x0=84. And 84 : 4 :: 231 : ^^--ll, the required number. 84. Reasoiv of the Rule.— It is evident that two num- bers, mnlUpaed or divided by the same, should produce pro- poi t.onatc results -It is otherwise, ho^yever, when tJie ^me quantity is added to, or subtracted froia tliem. Thus let tho given question be changed into the following. What number IS that wiuch being multiplied by 5, by 7, and by 9, the sum ol the products, plus 8, shall bo equal to 239 .> Assuming 4, the result will be 92. Then we cannot say 92 (84+8) : 4 : : 239 (231-|-8) : 11. For though 84 : 4 : : 231 : 11, it does not follow that h4-j-8 : 4 :: 2ol-j-8 : 11. Since, while [Sec. V. 29] we may multiply or divide the first and third terms of a geometrical proponiun by the same number, we cannot, without destroy- ing the proportion, add the same number to, or subtract it trom thorn. The question in this latt«r form belongs to tho rule of duuble pcsitiou. EXERCISES. 1. A teacher being asked how many pupils he had^ replied, if you add l, -] , and J- of the number together, the sum will be IS ; what was their number > Ans. 24. 2. What number is it, which, being increased by \, J , and i of itself, shall be 125 } Ans. 60. 3. A gentlcnuin distributed 78 pence among a num- ber of poor persons, consisting of men, women, and chil- dren ; to each man he gave GiL^ to each woman, 4rZ., and to each child, 2d. ; there were twice as many women as men, and three times as many children as women. How many were there of each } Ans. 3 men, f^ women, and IS children. ^ 4. A person bought a chaise, horse, and harness, for £t){) ; tbe horse came to twice the price of the harness, and the chaise to twice the price of the horse and har- POSITION. 347 flCKS. "What did he give for each ? Am. He gave for ihc harness, £C) I'Ss. 4(1. ; for the horse, i^^lS ijs. Sd. ; uud for the chaise, iD'lO. 5. A's age is double that of J5's ; IVs is trchlo that of C's ; and the sum of all their ages is 140. What is the age of each ? Ans. A's is S4, J3's 42, and C's 14. 6. After paying away J- of my money, and then } of the remainder, I had 72 guineas left. What had I at Krst ? yh^s. 120 guineas. 7. A can do a piece of work in 7 days ; J* can do the same in 5 days ; and C in G days. In what time will ull of them execute it ? Ans. in Ij^^ days. 8. A and B can do a piece of work in 10 days ; A by himself can do it in 15 days. In what time will 13 do it ? Ans\ In 30 days. 9. A cistern has three cocks ; when the first is opened all the water runs out in one hour ; when the second is opened, it runs out in two hours ; and when tlic tlurd i? opened, in three hours. In what time Avill it run out, it' ail the cocks are kept open together r Ans. In /y hours. 10. What is that number whose ■}, J-, and -} parts, taken together, make 27 ? Ans. 42. 11. There are 5 mills; the first grinds 7 bushels of corn in 1 hour, the second 5 in the same time, the third 4, the fourth 3, and the fifth 1. In what time will the five grind 500 bushels, if they work togctlier ? A71S. In 25 hours. 12. There is a cistern which can be filled by a cock in 12 hours ; it has another cock in the bottom, by which it can be emptied in IS hours. In what time will it be filled, if both are left open ? Ans. In 36 hours, 35. Doiibh Position . — Rule I. Assume two con- venient numbers, and pe.'form upon them the processes supposed by the question, marking the error derived from each with + or — , according as it is an error of e.iwvs, ©r of defect. Multiply each assumed number into the error which belongs to the other ; and, if the errors are hot/i plus, or hoth minus, divide the diJJ'crence of the products by tho difference, of the errors. ]3ut, if one is a plus, and the other is a minus error, divide the sum of I M 348 POSITION. the products by tlio mm of the errors. I,, either cas(^ the result will bo the nuiriber 60U'.-lit, or an apuroxi iiiatiuu to it. ^ ■ Exami'm: 1.— If t„ 4 timpg the price of my horse XIO U added, the sum will be £100. AVliat did it cast '? As.suming muubers wliich give two errors oi" excess-^ First, lot 28 be one of thoni, Multiply by 4 112 Add 10 From 122, the result obtained, subtract 100, the result required, and the remainder, +22, is an error of crrew. iMultiply by 31, the other assumsd uuinlwr and 082 will bo the product. Next,, let the assumed number be 31 Multiply by i 124 Add 10 From 134, the result obtained, subtract 100, the result required, , and the remainder, -{-34, is an error o?nrrc'^<i. Multiply by 28, the other assumed uum, and 'J52 will bo tlic produot. From this subtract G82, thoproductfouud.diove, divide by 12)270 and the required quantity br22r)=r£22 10s. Difference of crrora=34-.':;^12, the immber h\ whieli ._. .lift iiiimhdv ]\x «']ii,.1 we have divided. _ 36. Rkason of thk ?vUle.— AVhen in cxrimnle 1, we mul- tiply 28 and 81 by 4, we-iuuitiply the error belongini to e.icli by 4. Hence 122 and 134 are, reHpecLivelj, cquuf to tlie true result, plus 4 times one of the error.s. Subti-;ictiu<r tO(3, tb<j true result, from e;icli of them, we obtahi 22 (4 tiiue.ri >e errur in 28) and iJ4 (4 times tlie errur in 81). But, as numbers are propurtioniil to their '^y?//miltiples the error in 28 : the error in 31 : : 22 (a multiple of the for- mer) : 84 (au enuinuiltiple of the latter). And from the nature of proportion [Sec. V. 21]— POSITION. 349 The error in^28x34==thejBiwrM^^ Cut 682= tiio~error in"31-f t)ie required number X22. And 95'2=the error in 28-|-tiie required uuiabevX34. Or, since to multiply quantities under the vinculum [Sco [I. 84], we are to multiply each of them— ■ t)82=22 times the error in 3i-f-22 times the required number. it52=34 times the error in284-3i times the vequiied number. Subtracting the upper from the lower line, we shall have 952—682=34 times the error in 28—22 times the error in 31-4-84 times the required numbcv— 22 times the required number. • oq o i But, as -we have seen above, 34 times the error in jy=_'.i times the error in 31. Therefore, 34 time:s the error in 28—22 times the error in 31=0; that is, the two quantities cancel each other, and may be omitted. We shall then have 952 — 082=34 times the required number— 22 times the re- quired number; or 270=34-22 (=12) times the required number. And, [Sec. V. G] dividing both the equal quanti- ties by 12, OTA 34—22 "in- (22 '5) = - T^ *i"^C'* (once) the required number. 37. ExAiNiPi.E 2.— Using tho same example, and assivming nmnbora which ^ixa two errors of defect. Let them be 14, and 3.0 — 14 16 4 4 50 10 GO, the rosnlfc obtained, 100, the result required, - 34, an error of defect. 10 04 10 74, tiie result ol)tainod, 100, the resuk required, — 20, an error of defect. 14 544 304 304 Difference of errors : III •I : 34 — 20 = 8. 8)180 22-5 =£22 lOs., is the required quantity. In this example 34=.four times the error (of defect) in 14; and 26= four times the error (of detect) in IG. And, yineo ftiumbevs are pvoportioptd to iJieir equimultiples, The error in 14 : ih - error in IG : : 34 : 2(5. Therefore The error in ! lx2G=:tho error i u 16 X34. But 544=the required number — the error in 16X81 And3Gl=thc required number — the error in MX2G «i 2 1 POSITIOX. If wc subtract tho lower from tlic upper lino, we shall Imvo 644 — o(>4=(rcinuviiig the vinculuia, uud changing tho «igu [Sec. 11. It)]) ;M times the requiriMl number— 2(J times tTio lequired iiuuiber — ol liiiuis the civui- in 1G4--'J times tho error ill 14. IJut we found above that 34 times the error in 16=-'26 times tlie error in 14. Therefore — 34 times tho error in l<j, nnd4-"<> times tlie error in 14=i0, and niiiy be onntted. We will then have 544 — ;J(>1=34 times the required number — 2t> times the required number; or 180=8 time^ the required number; and, dividing botli these equal quantities by 8, 180 8 — Q- (22'5) =,-T times (once) tlie requtrcd number. 38. ExAMi'LE 3. — Using still the same example, and as- sinnlng numbers "vvhich will give an error of extaw, and an error of defect. Let thorn be 15, and 23 -- 15 4 60 10 70, tho rosiult ohto.intMl. 100, tho result required. - SO, an error of defect. 23 C'JO 30 23 4 92 10 102, tho result obtained. 100, tho result rcquirod. -|-2, an err 01' of exi-e^s. 15 30 Sum of errors = 30 -f- 2 : 32. 32)720 22-5 = £22 lO.s-., the required quantity. In this example 30 is 4 times the error (of defect) in 15; and 2, 4 times the error (of excess) in 23. And, since numbers are proportioned to the equimultiples, The error in 23 : the error in 15 : : 2 : 30. Therefore The error in 23X30=the error in 15X2. But l)UO=::the required iiumber-f-the error in 23x30. And 30=the required number — the error in 15X2. If Ave add these two linos together, we sliall have b!!0-f 30= (removi;ig tlie vinculum) 30 times the required numbor+ tvvice the required number -{-30 times the error in 23 — twice ihe error in 15. But we found above tliat 30 X the error in 23r=2v;;io error in 15. Therefore 30!i^the error in 23 --2 X the 'irror in 15=0. I'O'MTIO: 351 , S . w';*'>-> -t.-^- A'ud divldiug each «C tl..e equal quantities by 32. '^20,<,2.6^=- times (once) the required number. 8 ' oit The given questions might be changed into one belongmg to sin^'/e position, thus— p„u. times the price f "/.hj^'i =» °X l90.''"wlaf ..id or four times the price of "X ''<'''•« '''"j" „„ ^frort of the mind 39. ExAMPLK 4.-What is that number which is equal to 4 times its square root +21 '? Assume 64 on'' 81- 4 n2 21 ^81= 9 4 36 21 53, result obtainc'l. G4, result ro<iuired. 81 57, result obtained 81, result required 64 1536 891 13)645 The first approximation ia 49'6154 U is evident thatU and 24^ ^Z'lIX^ '^^^ „,„„bcrs multiplied or '^"'^'"LteuSe rule is founded, docs therefore, as the reason "P°" "™„o"hnation. Substituting ?,^iV >S;;^Wrfo?r or^-a-ssSeTnuntbers. we ohtatn . (Btill nearer approximation. TT T7m,l the errors by the last rule ; then kind), or their sum ^f they are o^J^"J^ ^ ^^^e of error which has been used as multiplier. 352 rosiTiox. KxAMrr.K. — Taking the Hamo an in tlic lust rule, and a/ Burning I'J and 23 as tho reiiuired numbor. 19 25 4 4 70 10 8(5 the result obtained. 100 the result required. —14, is error oi defect. 110 the result o))tainfMl. loo the result required. -f-10, ia error of excels. The errors are of different Y\x\({^; and their sum is 14-f ro=24 j and tho difference of the assumed numbers is 25 — 1U=(}. Therefore 14 one of the errors, is multiplied by 0, by the difference of the numbers. Then divide by 24)84 and 3-5 is the correction for 19, the number which gave an error of 14. 194-(the error being one of defcpt, the correction is to bo added) 3 5=22 5=£:i2 IDs. is tho required quantity. 41. Reason of the Rule. — Tlie diifercnce of tho results arising from tlie use of the different assumed numbers (tho difference of the errors) : the difference between tho result ob- tained b^'' using one of the assumed numbers and tliat obtained by using the true number (one of tlie errors) : : the difference between the numbers in the former case (the difference betweea the a.sHumed numbors) : the difference between thu numbei'S . in the latter case (tlie difference between the true mmiber, and that arjsumed number wiiich produced tlie erior placed in the thir<l term — tliat is the correction required by that assumed number). It is clear that the difference between the numbers used produces a proportional difference in the results. For tlie results are different, only because tho difference between the assumed numbers has been multipliccl, or divided, or both — iu acconlance with tho conditions of the question. Thus, in the present iuytsmce, 25 pl'oduoes a greater result thau 19, because 0, the difference becween 19 and 25, has been multi- plied by 4. For 25x4=s=19x4-f-6x4. And it is this 6X4 which makes up 24, the rtal difference of tlie errors. — The difference between a negative and positive result being the sum of the differences between each of them and no result. Tbus, if I gain 10s., 1 am richer to the amount of 24*. than if 1 lose li.*. ^ rosiTiuN. 353 ilo, and M t o))tainfMl. fc required. of execs'^. im is 14+ era is 25 — ors. Then do number on 13 to bo the results iiubers (tlio 3 residt ob- lat obtained e difference ace between \ii numbers lumber, and aced in the at assumed mbpvs used s. For tlie )etween the , or both — . Tims, in ;lt thau 19, been multi- is this 6x4 rrors. — The . being the I no result. 24«. than if t EXEUCISES. 13. What number is it whit;li, boiur; niultipliod by 3, Uio product \n\\\\% increasod by 4, and the sum divided by 8, the quotieut will bo \V2 ? Ans. 84. 14. A4!ron asked liis fatlier how old lie was, and re- ceived the foUowin,-]; answer. Your ago is now J- of What are their mine, but 5 years ago it was only i. ag(\s } Ans. 8;) and 20 ' 15. A workman was hired for 30 days at 2s. Qd. for every day ho worked, but with this condition, thafc^ for every day he did not work, he should forfeit a shilling. At the end of the- time he received £,2 14^., how many days did he work ? Am. 24. 16. llcquired what number it is from which, if 34 be taken, 3 times the remainder will exceed it by \ o^ itself .= Ans. 58-=. . 17. A and 13 go out of a town by the same road. A g(jcis 8 miles each day ; 15 goes 1 mile the fir.st day, 2 the second, 3 ' ' ' ' take A } the third, &c. When will B over- Suppose A. 5 8 40 15 B. 1 2 3 5 Suppose A. 7 8 50 28 5)25 Is --a 7 35 20 1)15 7)28 -4 5 20 B. 1 2 • 1 O 4 5 G 7 28 5 4=1 "Wc divide tho ciilive eri'nr by the number of daya iu each c.-.sc, -which gives tlie error iu one day. 18. A gentleman hires two labourers; to the one bo gives M. each day; to the other, on the first day, 2(/., on the second day, Ad.^ on the third d;iy, 6^/., &.c. In how many days will they earn :tn e'pinl sum } Aim. In 8. 11). What" are tho.s:j numhers whieh, when added, ij S54 I'OSrTION. make 25 ; but when ono in halved and the other douhled, give (H[na[ results ? Ans. 20 aud 5. 20. Two coutractoi-H, A and ]J, arc each to hiiild a wall of equal dimcusioiis ; A (iniploys as many men as finish 22J perches hi a r'ay ; Ij employs the first day as many as finish G perches, the second as many as finish 9, tlie third as many as finish 12, &c. In what time will they have built an e(|ual number of perches ? Ans. In 12 days. 21. What is vhat number whose ^, i, and -}, multi- plied together, make 24 ? Sujtposo 12 1=3 rroduct=i8 3 41 81 result obtained. 2-1 result rc(iuircd. +57 04, the cube of 4. 3648, product. 57+21=78 Suppose 4 Product=r;2 a — U 3 result obtained. 24 result required. 1728, the cube of 12. 30288 To this pruduct 3018 is added. 5< 7-21=78. 78 )391)3 is the sum. And 512 tlio quotient. -3/512=8, is the required number. We nmltiply the alternate error by the cube of tlie supposed tmmbcr, because tlui errors belong to the g'^th part of the cube of tlie assumed numbers, and not to tlic nuinbei'S tlicniselvos ; for, in reality, it is the cube of some number that is required —tt'ince, 8 being hssumed, according to the question we have 22. What number is it whose 1, J-, ]-, and 1, multi- plied together, will produce 699S| .'^ Am. 36.^ 23. A said to B, give me one of your shillings, and I shall have twice as many as you will have left. B answered, if you give me Is., I shall have as many as you. Ilow ninny bird each } Avs. A 7, and B 5. POSITION. 355 24. There are two nuinbors wliich, when ailuo(1 to* gather, iiialca 30; but the J, J, and j, of the greater arv cqijal to |, a^ and |^, of the lesser. What are they ? Ans. Vj and Ijs. 2' A f^ontlomaii has 2 horsoa and a sacMlo worth j£50. Tbo ,s:iddlo, it' set ' n th* baek of the first linrse, will make his value doubic that of the second ; but if set on the baek of the second horse, it will make his value treble that of the first. AVhat is the value of each horse ? j v. £30 and iJlO. 2C>. A gentleman finding sov ral ben;£^ars at his door, pavt! to eii;h 4d. and had Gd. left, but if he had given ikl. to each, he would have had 12d. too little. How many bop:gars were thure ? Aiis. 9. It is so likely tliat those ) are desirous of stud^inr; this subject further will be acquainted with the method of troatin<5 algebraic equations — which in miiiy case? nffords a so much simpler and easier mode of solvin,'; qu.;stions belonging to position — that we do not deem it necessary to enter further into it. QUESTrONS. 1. What is the diiTerence between single and double position.? [32]. 2. In what cases may we expect an exact answer by ihesc rules r [32 j . 3. Whtit is the rule for sin<>-le position ? [33] . 4. AVhat are the rules for double position .'* [35 and 40 j. MISCELLANEOUS EXERCLSES. 1. A father being disked by his son how old he was; voplied, your age is now ^ of mine ; but 4 years ago \i was only ^ of what mine is now ; what is the ago of each } A as. 70 and 14. 2. Find two numlxn-s, the dilForence of which is 30, nnd the relation between them as 7^ is to 3^.'' Am. 58 and 28. 3. Find two numbers whose sum and product are equal, neither of (hem being 2 ' Ans. 10 and 1^. .% o.. \t^ IMAGE EVALUATION TEST TARGET (MT-3) /. ^° 'C^x - ^> /^^^ j/^ C^, :/. % 1.0 I.I 11.25 1.4 IM [2.0 1.6 PhotograDhic Sciences Corporation 23 WEST MAIN STREET WEBSTER, NY. 14580 (716) 872-4503 ,,-y^': o 356 EXERCISES. What is the t;um of tlie series -J-, i, }, &c. ? Ans. 1. 4. A^porson being asked the hour of tlio day, answered, It is between 5 and G, and 1;otli the hour and minuto hands are together. lle(iuired what it was ? Ana. 27 f\ minutes past 5. 5. 6. What is the sum of the .series' |,'y\, j%y j-'A) ^^- • Ans. If 7. A person had a salary of £75 a year, and let it remain unpaid for 17 years. How much had he to receive at the end of that time, allowing 6 per cent, per annum compound mtercst, payable half-yearly ? Ans. £204 17s. lO^d. 8. Divide 20 into two such pav^s as that, when tho" greater is divided by the less, and the loss by tlie greater, and the greater quotient is multiplied ))y 4, and the less by 64, the products shall be equal.? Ans. 4 and IG. 9. Divide 21 into two such parts, as that when the less is divided by the greater, and the greater by the loi'S, and the greater quotient is multiplied by 5, and ih" less by 125, the products shall be e(pial ? Ans. 3'- and 171. 1" A, B, and C, can finish a piece of work in 10 days; }> and C will do it in 16 days. In what time will A do it by himself? Ans. 26| days. 1. A can trench a garden in 10 days, B in 12, and in 14. In what time will it be done by the three if thoy work together ? Ams. In 3-,-Yt ^'-^J^- 12. What number is it which, divided by 16, will leave 3 ; but which, divided by 9, will leave 4 ? Aiis. 67 i3. What number is it which, divided by 7, will leave 4; but divided by 4, will leave 2 ? Ans. IS. 14. If £100, put to interest at a certain rate, wih, at the end of 3 years, be augmented to £115'7G25 (compound hitercst being allowed), what principal and interest will hi due at the end of the first year ? Ans. £105. 15. An elderly person in trade, desirous of a little respite, pi'oposcs to admit a sober, and industrious young person to a share in the business ; and to encourage him, ho olfors, that if hi^ circumstances allow him to F,XERCrSE!5. 357 advance £100, his salary shall be £40 a year ; that if ho is able to advance £200, he shall have £55 ; but that it he can advance £300, he shall receive £70 annually. In this proposal, what was allowed for his attendance simply ? Am. £25 a year. 16. If 6 apples and 7 pears cost 33 pence, and 10 apples and 8 poars 44 pence, what is the price of one apple and one pear .?* Ans. 2d. is the price of an apple, and 3d. of a pear. 17. Find three such numbers as that the first and I the sum of the other two, the second and i the sum of the other two, the third and \ the sum of the other two will make 34 ? A^is. 10, 22, 26. 18. Find a number, to which, if you add 1, the sum will be d"visible by 3 ; but if you add 3, the sum will be divLsiVie by 4 ^ Am. 17. 19. A market woman bought a certain number of eggs, at two a penny, and r,s many more at 3 a penny ; and having sold them all at the rate of five for 2^., she found she had lost fourpence. How many eggs did she buy .? Am. 240. 20. A person was desirous of giving 3d. a piece to some beggars, but found he had 8^. too little ; he there- fore gave each of them 2d., and had then Sd. remain- llequired the number of beggars.? Am. 11. 21. A servant agreed to live with his master for £8 a year, and a suit of clothes. But being turned out at the end of 7 months, he received only £2 135. 4d. and the suit of clothes ; what was its value } Am, _ 16.9. 22. There is a number, consisting of two places of figures, which is equal to four times the sum of its dtgits, and if 18 be added to it, its digits will be in- verted. Wiiat is the number ? Am. 24. 23. Divide the number 10 into three such parts, that if the first is multiplied by 2, the second by 3, and the third by 4, the three products will be equal .? Am. 24. Divide the number 90 into four such parts that, If the first i;^ increased by 2, the second dhninished by 2, the third multipli(!d by 2, and the fourth divided by mg 358 EXERCISES. % 2, the sum, clIfForcncc, product, and quotient will bo equal : Ans. 18, 22, 10, 40. 25. \Vliat fraction is that, to the numerator of which, if ] is added, its viluo will be i ; but if 1 bo added to tHe denominator, its value will be •} ? Ans. j%. 2^3. 21 gallons were drawn out of a cask of wine, which had leaked away a third part, and the cask being then guaged, was found to be half full. How much did it hold ? Ans. 126 gallons. 27. There is a number, ^ of which, being divided by 6, I of it by 4, and J- of 'it by 3, each quotient will be 9 ? Ans. 108. 28. Having counted my books, I found that when I multiplied together i, j, and f of their number, the product was 162000. How many had I ? Ans. 120. 29. Find the sum of the series l+'^-f j + |, &o. .? Ans. 2. 30. A can build a wall in 12 days, by getting 2 days' assistance from B ; and B can build it in 8 days, by getting 4 days' assistance from A. In what time will both together build it ? Ans. In 6f days. 31. A and B can perform a pisce of work in 8 days, when the days are 12 hours long ; A, by himself, can do it in 12 days, of 16 hours each. In how many days of 14 liours long will B do it } Ans. 13^. 32. in a mixture of spirits and water, | of the whole plus 25 gallons was spirits, but i of the whole minus 5. gallons was water. How many gallons were there of each } Ans. 85 of spirits, and 35 of water. 33. A person passed } of his age in childhood, yV of it in youth, | of it +5 years in matrimony ; he had then a son whom he survived 4 years, and who reached only i the age of his father. At what age did this per- son die ? Ans. At the age of 84. 34. What number is that whose i ejjcecds its \ by 72 ? Ans. 540. ^ _ . 35. A vintner has a vessel of wine containing 500 gallons ; drawing 50 gallons, he tlicn fills up the cask with water. After doing this five times, how much wine and how much water are in the cask.? Ans 295^j)_ gallons of wine, and 204 J |i gallons of water. '' ! , ni EXERCISES. 350 45. A mother an<l two daugliters working together nil 3 lb of fliix in one day ; the mother, by herself, ian do it in 2i days ; and the eklest daughter m ^j days. In what time can the youngest do it.? Ans. In f')— davs. 37^ A merchant loads two vessels, A and B ; into A he puts 150 hogsheads of wine, and into B 240 hogs- heads. The ships, having to pny toll, A gives 1 hogs- 15 <.;ivos 1 hogshead and 3().s\ iich hogshead valued ? head, and receives V2s. besides. At how much was c Ans. £4. 12.S-. , ^, . 38. Tlireo merchants traffic in company, and their stock is i2400 ; the money of A continued in trade 5 months, that of B six months, and that of G nine months; and they gained £^7b, which they divided cfpially. What stock did each put in.? Ans. AiilbT^-j, 39. A fonntain has 4 cocks, A, B, C, and D,_and unilcr it stands a cistern, which can be filled by A lu G, by B in 8, by in 10, and by D in * .- hours ; the cistern has 4 cocks, E, F, 0, and II; and can be emptied by E in G, by F in 5, by Q in 4, and by II m 3 hours. "Suppose the ci^^tern is full oi water, and that the S cocks are all open, in what time will it be emptied ? Ans. In2,^g hours. 40. What is the value of -2^07' ? Ans. if 41 What is the value of -541 G' ? Ans. Yi- 42. What is the value of •0^7G923' t Ans. yV- 43. There are" three fishermen. A, B, and C, who have each caught a certain number of fish ; when A's fish and B'sare put to:rother, they make 110 ; when B's and CVs are put together, they make 130; and when A's and C's are put together, they make 120. It the fish is divided equally among them, what will be each mairs share; and how many fish did each of them catch ? Ans. l<lach man had GO lor his share ; A caught 50, B GO, and G 70. 44. There is a golden cup valued at 70 crowns, and two heaps of crowns. The cup and first heap, are wortli 4 times the value of the second heap ; but the cup and second heap, ai-e worth double the value of the first 3r,o EXERCISES. heap. ITow many crowns arc there in eacli lieap ? Ana oO ill one, and 30 in another. 45. A certain number of horse and foot soldiers ai'O to be ferried over a river ; and tliey agree to pay 2^d. for two horse, and Sid. for seven foot soldiers ; seven foot always followed two horse soldiers ; and when they were 'dl over, the ferryman received ^£25. How many horse and foot soldiers were there ? Ans. 2000 horse, and 7000 foot. 46. The hour and minute hands of a watch arc' to- gether at 12 ; when will they be together again ? Ans. at 5/,- minutes past 1 o'clock. 47. A and 13 are at opposite sides of a wood 135 fathoms in compass. They begin to go round it, in the Same direction, and a* the same time ; A goes at the rate of 1 1 ftithoms in 2 minutes, and B at that of 17 in 3 minutes. How many rounds will each make, before one overtakes the other ? Ans. A wiU go 17, and 13 16J-. 48. A, B, and 0, start at the same time, from the (same point, and in tlie same direction, round an island 73 miles in circumference ; A goes at the rate of 6, B at the rate of 10, and C at the rate of 16 miles per day ^ In what time will they be all together again .^ Ans. in 36|- days IX M-ATIIEMATICAL TABLES LOGATUTIIMS OF NUMBERS FllOM 1 TO 10,000, WITH DIFFKIIENCES AND PROPOllTIONAL I'AllTS. I ' 1 m^^M Numbers f •om 1 to 100. # I No. Log. j No. Log. NO. Log. No. Log. No. Log. I 1 . O-O'JOOOO •21 1-32'2'219 41 1-614784 61 1-785330 81 1-908486 H ■i 0- 301030 •22 1-342443 44 1-043249 64 1-792394 84 1-913314 1 3 0-477141 -23 1-361743 13 I -0334 .3 63 1-709341 83 1-919073 4 0-60-JOGO 21 1-380211 U 1-C43453 64 1-800180 34 1-9-24479 1 (i 0- 690970 45 '26 1-397040 45 46 1-053413 05 1-814913 35 1-9-29419 !jHH 0'778151 1-414973 1-664758 66 1-819544 66 1-9^4498 H 0-Sl.JOOJ •27 1-431364 47 1-674093 67 1-346075 87 1 9.39519 ^H 8 0-903000 •23 1-417158 43 1- 631-241 6S 1-834509 83 1-944483 mH 9 0-9J4243 29 1 •462393 49 1-6901% 69 1-838349 89 1-949390 ■ 10 1-000000 30 31 1-4771-21 60 51 1-693970 70 1-845098 90 1-954413 ■ 11 l-0413ri3 1-491364 1-707570 71 1-851'253 91 1-959041 M 12 1 -079181 32 1-505150 53 1-716003 74 1-8573.34 92 1-963783 ^H 13 1-113943 33 1-518514 63 1-7-24476 73 1-863343 93 1 -968433 ^H 14 l-UOlia 31 1-531479 54 1-732394 74 1-869434 94 1-973128 iHI 15 1- 170091 35 36 1-644063 65 56 !■ 740303 1 75 1-875061 95 1-977724 ^H IG l--2041-iO 1-550303 1-748188 76 1 -830314 96 1-98'2471 m 17 1 --230449 37 1-533404 57 1 -766875 77 1-886491 97 1-986772 H 13 \-2iio-2l'i 33 1 •5r9784 53 1-763143 73 1-894095 93 1-991226 ■ 19 1-478754 39 1-591005 59 1-770854 79 1 -SO? 647 99 1-995635 ■ •2ii I • 301030 40 1-604060 CO 1-773151 SO 1 -903090 100 2-000000 IB^^I "mm 302 rOGARITIIMP. r 1' 100 1 1 3 1 3 1 4 6 6 7 1 8 1). 132 :oouuoooooi3i 0008O.sOOI3ni|0017.34 UO'2IOO 002.598'00.3029 00.3461 003891 II 11 in-Jll 4751 6181 660!)! fif38 6466 68941 7321 1 7748 8171 128 Hii •2 K000[ !)0-20 3'01'JdH7 013'20!) 9l,-.l !)n76 010300 010724 011147i01l570011993;0l211.. 121 101 tU30KO 014100 4.V21 4910 6360 5779! 6197 6616 120 Ifiti I 7033 74')] 7868 8284 8700 9116 9532 9947 020361 020775 116 ■-'o; 6 0-2llH!)uai(i03 022ai6]02'2428 022811 023252 023604 024075 4486 4896 112 VilH G TMHy 571;-, 61 •2.-. 6533! f:942 7350 77.-)7 8164 8571 '3978 108 •J!M) 7 '3' ll 978l'i 03019.) 030(ilU):031004|031 108 031812 03-2216 032619 0M02I 101 ;-t;il /24',mi26 42;;7 4628, 6029 5430 6830 62301 6629 702.M !0l» ;);:) 110 ',4:20 78-26 8-223 H620| 9017 9414 9811 040207,040602 040998 ;J97 .193 (i;i3:)3 011787 042l82|012.-)76 042909 043302 043755 044148 044.540 0449.32 ;j^ 1 ;':.3-23j 6714 OlOril 6I9.-)I 688.'. 7275 7064 8053 8442 8830 390 7li • 1 9218 iUiOli 9993 J0380'n.'.070fi 051153 051538 051924 0.52309 0.5-2(i91 386 li:i 3 003('7S (l;-(3403 053846 4230 461.1 4;)96 5378 ,OT60 6112 6521 .■i83 1.^1 4 UDO/)! 7-2b0 7006 8046 8126 8805 9IH5 9563 9942 060320 .179 iy:< t OGO()!I800107') Oai4.>2 061829 062200 0(!2J82 002958 063333 063709 4083 376 •M^ 6 4i;)S 483-2 520(1 ooSn 6!»53| 6320 6699 7071 1 7443 7815 373 •]«.-, 7 81Si)i 8;)i")7 8928 9298 UtiUS 07003S 070407 070776,071115 071611 370 aoo H 0718+2 07U^2;")0 072617 07298.-, 0733..2 3718 4085 4151 4816 5182 366 :M() lyo .');:.47 .01)12 6276 0640 7004 7308 7731 8094 8457 8819 163 :J60 079181 07l)")43 0799U4;080200'080ll26 080937 081347 081707 082067 08212(; 3;-. 1 08278.-., 083141 083.-.03 3861 4219 4576 493 1 5291 5647 6001 357 70 o 0;l!iiH 0710 707 1 712{i 7781 8136 8490 88 15 9198 9552 355 101 3 990;V090-208 09061 1]090903 091315 09 I6(i7 09-2018 092370 092721 093071 :J52 i;i<i 4 093t-2-2J 3772 4122 4471 4-20 5169 5513 rySGii 6215 656^ .549 17-1 6 091(1 ~-2:u 7604 79.-,l 8298 864 I 8990 9335 9681 10002(i 346 20;i 100371' 10071.". 1010;;9 10! 103 101717 10'2091 10-2434 102777 103119 3162 .113 ■211 7 3804 4140 4 187 482 S 6169 6510 5851 6191 6531 6871 :t41 •J7m H 7210 7649 78.'-8 8-227 8565 8903 9241 9,)79 9916 110253 138 ■M.i 9 110o90,1109'2G 111203 lll.)99|111934J112-270 112605 U2940 113275 3609 335 433 130 I!3s;i3;n4^277 114611 U4')44;ll;V278'lli)^ir, 115943 116-276 116608 11 694 f) ;;o 1 727 ll 7003 793; 8265 8595 8926 9256 9586 9915 120-24.-i 330 01 •2 1-20.-.74' 1-20903 1-2I-23I 1215(i0 121888 122216 122544 122871 123103 3525 ■',-lS !»; A 3^iV2 4178 4.)0t 4830 5150 6481 5806 6131 6451' 6781 325 i-ii) 4 710.-. 71^vft 77 -.3 8070 8399 8722 9045 9368 9690 130012 3-23 101 .'. 1.30334 130tJ.>;) 130977113^298 131619 131939 132260 132580! 132900 3219 .!21 IPS 3r.39 3858 4I77| 41;)o 0721 7037 7354I 7071 48141 01.33 5451 6709 6086 6403 .(18 ■s-i-') 1 7987 1 8303 8618 8934 9219 9561 .il6 \!..b 8 9879 140191 140.-;03 140822 141136 141450 141703 142076'142389 14270-2 :il i!)U 9 11301.) 3327 3639 3951 42631 4571 4835 6196 5507 5811- ill :!09 llu 140r23 140 138 14074S| 147058 147.3671147670 147985 148-294 148603 148911 30 1 9^2 1 9 9rv27 9835|150112|I50119;1507j6|151063;151.370;1o1676;15198-. :(07 (M o \ry2-i^» l;V2r>94 152900 3205 3510 3815 4120 4421 4728 .503'2 .(05 !)l) «J ;)330 5040 5943 C2I0 0549 6855 7154 74.57 7759 8061 .'.03 IJO 4 830^2 8004 890--) 92()0 9567 9868 160108 160469 160769 16106b :!ni 1 oO -• 101308 161607 161967 162200 162564:162863 3161 3460 37.58 4055 299 180 43"<3 4Q,jO 4947 5214 5541 6838 61.34 6430 (i726 702^- 297 ■J 10 7 7317 7013 7908 8203 8-197 8792 9086 93 SO 9674 990; 295 210 fi 170202 170.-i,-)0;i70818 171141 171434 171726 172019:17231 11172603 172895 J93 ■270 9 i;.o 3180| 3478 3709 4000 4351 4641 4932 6222 6512 680-j .'91 2.S9 170091 170381 Il700ro' 170969 177^i4.9 177536:177.825' 17811 3 178 lul 1786,Si' •js 1 8977! 9-2041 9-;.V2l !i839;i801-2t):i80413:i80690'l809SG 181-27-2'l816.>i- 287 c')0 ■) I.-'I814!I82129 182415:182700 2985 32701 .3555 3!s39 4123 440-. .'85 81 3 40.1)1 497 J 5259 6542 6S25 61(»8 0391 0674 6956 723f' .■83 irj 4 7.-.21' 7803 .808 1 83001 8647 89^18 9209 9490 9771 190051 281 110 ;') iyU33-i;lH(l()l:» 190892.191171 191451 !l917.30192010 192289:192507 28 li: ;79 llirf 312..| 340.; 308 1 3>)5;i •1237 4,-.ll 4792! 5.)(;9 5313 5i;-2: .'78 l'.)u 7 .'.900 6170 6453! 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3 8421 8468 8616 48 43 9 s)IO b)6l 969U41 8612 S6.^9j 87071 8706 8803 88,i0 8898 8946 8994 48 48 969089 9,>91.37 969186 969232 959280 959328'969376 969423 9.59471 5 1 9,. 18 9>fl6 9614 9661 9709 97.67 9804 9862 9900 99.17 48 9 2 i'ni'.)!} 960042 969090 9001.33 960186 960233 960281 960328 960376 960423 48 14 3 960171 0,-il8 0666 0613 0661 0709 0766 0804 0861 0899 48 19 4 0916 099 I 10-11 1089 1136 1184 1231 1279 1326 1.374 47 24 6 1121 1469 1616 1663 iOll 16.-.8 1706 1763 1801 1348 47 28 6 1896 1943 1990 2038 2'i86 2132 2180 2227 2276 2.322 4'r 33 7 2369 2417 2464 2611 2669 2606 2663 2701 2748 2795 47 38 8 2813 2890 2937 293;) 3032 3079 3126 3174 3221 3268 47 42 9 9;;() 3310 33G3 3110 34,67 3.601 3652 3599 3616 3693 3741 47 963788 963836 903882 963929 963977 964021 961071 961118 964166 961212 47 £ 1 ■1 JOO 4307 4304 4401 4148 4496 4612 4690 4637 4681 47 S 2 4731 4778 4826 4872 4919 ,1966 6013 6001 6103 6 1 56 47 i 14 3 6202 6249 6296 6313 6390 6137 64,84 6631 6.678 6626 47 19 4 .0672 6719 6766 6813 5860 5907 6964 6001 6048 6096 47 23 6 6142 0189 6236 6283 6329 6376 6423 6170 6617 66('4 47 29 6 O.ill t;6,i8 6706 0762 6799 6346 6892 6939 6986 7033 47 . 33 7 7080 7127 V173 72-,;0 7267 7314 7361 7408 71,5! 7.501 47 38 8 7,348 7,i96 7642 7 688 7736 7782, 78291 7876 7922 7969 47 42 9 80 1 6 S,)62 8109 8166 8203 824:-){ 8296; 8313 8390 8436 47 9684831968630 96,'^,;76 968623 968670|96-i7i6'968763 96881 968866 968903 47 6 1 MV,0 8996 9043 9090 OUlol 9183 9229 9276 9323 9369 47 9 2 9116 9163 9609 9660 9602 9649 9696 9742 9789 9836 47 14 3 9382 9928 9976 970021 970068 97011 ! 970161970207 970264 970300 47 io 4 970347 970393 970440 0486 Ou.iH (',y/9 06'j6; 0672 0719 0766 46 29 5 0812 0368 0904 0951 0997 1044 1090 1137 1183 1229 40 23 6 1276 1.322 1369 1416 1161 1.608 1. 6.541 1601 1647 1693 46 32 7 1740 1786 1332 1879 1926 1971 2018 2064 2110 2167 46 37 8 2203 2249 2296 2342 2388 2134 2481 2527 2.573 2619 46 41 9 2666 271-2 2768 2804 2861 2397 2943 2989 3035 30t.2 46 !J ||i i! iil iii^ . . i 1 ,t I J76 LOOARITHMS. 5 14 IH 37 41 6 9 14 18 2:» •i7 3-2 3t< 41 N. 340 1 2 3 4 900 J 3 4 6 6 7 9731-iS 30D0 4051 451i 4972 f)432 5891 63f)0 fisoa 72aa 1)77724 8181 8037 9093 9548 980003 0458 0912 13fifi 1819 1 973174 3()36 4097 j 45581 60181 6478 6937 6396 6854 7312 973220 3f.82 4143 4604 6064 6524 6983 6442 6900 7358 973266 3728 4189 4660 6110 5570 6029 6488 6946 7403 977769 8926 86u3 9138 9594 980049 0503 0957 1411 1864 977815 8272 8728 9184 9639 980094 0549 1003 1456 1909 977861 8317 8774 9230 9685 980140 0694 1048 1501 1954 973313 3771 4235 4696 5156 5616 6076 6533 6992 7449 977906 8363 8819 9275 9730 980185 0610 1093 164' 2000 977952 0409 8865 9321 9776 980231 0685 1139 1592 2045 977998 8454 8911 9366 9821 980276 0730 1184 1637 2090 982452 2904 3356 3307 4257 4707 6157 6606 6055 6503 982497 2949 3401 3852 4302 4752 6202 6651 6100 6548 8 9 973369,973405 97345 1 973497 3820 3866 3913 3959 4281 4327 4374 4420 4742 47 88 4834 4880 5202 6218 6294 6340 6062 6707 6763 6799 6121 6167 6212 6258 6579 6626 6671 6717 7037 7083 7129 7175 7495 7541 7686 7632 978043 8500 8956 9412 9867 980322 0776 1229 1683 2136 973543 4005 4166 4926 6386 5845 6304 6763 7'?20 7678 978089 8546 9002 9457 9912 980307 0821 1275 1728 2181 9SG951 7398 7845 8291 8737 91 S3 9628 |990072 j 0516 0960 982543 2994 3446 3897 4347 4797 6247 5696 6144 6593 982583 3040 3491 3942 4392 4812 6292 5741 6189 6637 982633 3085 3536 3987 4137 4887 8337 6786 6234 668i 978136 8591 9047 9503 9968 980412 0867 1320 1773 3226 982678 3130 3581 4032 4482 4932 6382 6830 6279 6727 936996 987040; 7443 7890 j 8330 8782 9227 9072 990117 0561 1004 748S 7934 b;>jl (■^S20 9272 9717 990UU 0605 1049 9870S5 753« 7979 8425 837 1 9316 9761 990206 0650 1093 1802 1846 1890 2244 2283 2333 2686 2730 2774 3127 3172 3216 3568 3613 3657 4009 4053 4097 4449 4493 4537 43B9 4933 4977 5328 6372 5416 937130 7577 8024 8470 8916 9361 9306 990250 0694 1137 45 45 46 45 45 45 45 45 45 45 937176 7622 8068 8514 8960 9406 9360 990294 0738 118! 5 991359 iiOl 103,991448991492 1935 2377 2819 3200 3701 4141 4531 5021 5460 67 995811 995854 995898 6337 6774 7212 6205 6249 6293 6643 6637 6731 7030 7124 7168 7517 7501 7005 7954 7998 8041 8390 8434 8477 8326 B869 8913 9261 9305 9348 9096 9739 9783 8086 8521 8956 9392 0826 991536 1979 2421 2363 3304 3745 4185 4625 5065 5504 995942 6380 6818 7255 7C:)2 8129 8564 9000 9435 9870 991530 2023 2165 200 3343 3739 4229 4669 6103 654' 1 995986 6424 6662 7299 7"36 8172 8608 9043 9479 99i8 45 45 45 45 45 45 44 44 44 44 991626 2067 2609 2901 3392 3833 4273 4713 6152 6591 996030 6468 6906 7343 7779 8216 8652 903- 9522 9967 44 44 44 44 •11 44 44 44 44 43 A TABLE OF S(iUAHKS, tj0hK9, AND ROOTS. 377 1 9 D. n 73.543 46 4005 46 41t)b 46 4926 46 6386 46 5846 46 6304 46 6763 46 7'?20 46 7678 46 i )78136 46 8591 46 > 9047 46 9503 46 I 9908 46 r < )80412 46 0867 46 > 1320 45 i 1773 46 1 3226 46 J 982678 45 5 3130 46 i 3581 46 ? 4032 45 7 4482 46 7 4932 46 7 6382 45 6 6930 46 4 6.i79 45 2 6727 46 987176 45 7 7622 45 4 8068 45 (» 8514 48 6 8960 45 1 9406 45 6 9360 44 990294 44 4 0738 44 7 1182 44 to 991626 44 '3 2067 44 >5 260S 44 )7 2901 44 H 3393 44 3.0 3833 44 29 427f 44 jt) 47K 44 )H 61oV 44 t7 5591 44 it 99603( 44 i<l 646t 44 j'J 690( 44 3£ ?34; 44 It: ; 1 r •ii ?v . 821( 44 Bt i 866- 44 4J 1 908" 44 7i t 952- i 44 li t 996' r 43 No. flqonre. Cut. p. Sq. Root. Cuho Iloot No. 64 Sqiiaru C'tilie. .^q. Root. Cube Root I 1 1 1-0000000 1 000000 4096 262144 8-0000001) 4-000000 a 4 8 1-4142136 1-260021 66 4-226 274626 8-06-2-2677 1-020726 3 9 27 l-7320.')08 1-412260 66 4366 287496 8-1240331 4-041240 4 16 64 2-0000000 1-687401 "B7 4489 300763 8-1863628 4-0616-18 6 25 125 2-2360680 1 -709976 68 4624 314432 8-2462113 4-081656 6 36 216 2-4494897 1-817121 69 4«61 328609 8-2066-239 4-101660 7 49 313 2-6467613 1-912931 70 4^)00 343000 8-3666003 1-121286 8 64 512 2-8284271 i- 000000 71 6041 367911 8-4-261498 4-140318 9 81 729 3-0000000 2-080084 72 6184 373248 8-48.5-2814 1-160168 .. 10 100 1000 3-102-2777 2-164136 73 6329 389017 8-6440037 4-179339 11 121 1331 3-3106-248 2-223980 74 547<) 40.5224 8-60-23-263 4-193336 12 144 172S 3-4011016 2-289128 76 5626 421876 8-6602640 1-217163 13 169 2197 3 -6066513 2 -361. 3;<6 76 6776 438976 8-7177979 4--i368J4 14 196 2744 3-7410574 2-410142 77 6929 466633 8-7749644 4-254321 16 226 3376 3-87-29833 2 -466212 73 6084 474562 8-8317609 ■1-272669 16 266 4096 4-0000000 2-51984-2 79 6-241 493039 8-0881944 4-290841 17 289 4913 4-1231066 2-571282 80 6100 61-2000 8-944-2719 4-308870 IS S54 6832 4-2420407'2-62'>741 81 6661 631411 9-0000000 4 -.326749 19 861 6859 4-3688939 2-608402 82 67-24 661368 9-0663861 4-344481 20 400 8000 4-4721300 2-714418 83 6889 671787 9-11043.36 4-36-2071 21 441 9201 4-6826767,2-763924 84 8,^ 7066 692704 9-1661614 4-379619 22 484 10048 4- 09041 58'2- 80-3039 7226 014126 9-219.6416 4 -.396830 23 629 12167 4-796831612-843867 86 7396 636066 9-2736186 4-414006 24 676 13824 4 -8989796 ,2 -334499 8T 7669 668603 9-3-27.3791 4-431047 26 026 15626 &-0000000!2-9-24018 83 7744 681472 9-380.-3316 4-447960 26 676 17676 6-0990196;2-9d2496 y:) 7921 704969 9-4339811 4-464746 27 729 19633 5-19615-24 3-000000 90 8100 7-29000 9-436.3330 4-481406 20 784 21962 6 -29160-26'3- 036689 91 8281 7.63671 9-6.393920 4-497941 29 841 24389 6-3361648 3-072.'il7 92 8461 778688 9 -.69 16630 4-514367 30 900 27000 6-4772266 3-107'232 93 8619 804367 9-6436608 4-530656 31 961 29791 6-6677644 3- 141.381 94 S336 830584 9-6953697 4-6468.36 32 102{ 32768 6-6668612 3-174802 96 9026 867376 9-7467943 4-662903 ■ 33 1089 36937 5 -74 166-26 3- -207634 96 9216 884730 9-7979.590 4-578867 ! 34 1166 39304 6-8309619 3-23961-J 97 9409 912073 9-8488678 1-594701 85 1226 4287;-. 6-9160798 3-271006 93 9604 94119-2 9-8991949 4-610436 .S6 1296 46666 6-0000000:3-3019-27 99 9801 970-299 9-949374 1 4 - 626066 .Hi 1361) bmr,.\ 6 -0327026;3- 3322-22 100 1 0000 1000000 10-0000000 4-641689 3S 1444 64872 0-1614140,3-30197;> 101 10201 1030301 10-0498766 4-667010 :!0 1621 69319 6- 244 9980:3-39 1211 102 10401 1061208 10-0996019!4-67-2329 40 KiOO 64000 6 -32 15663 3-419962 103 10609 1092727 10-1488910 4-687643 ■!1 Hi 3 1 68921 6-4031242:3-418217 104 10:ilt) 1124861 10- 19.303904 -702669 42 1764 7408d 6-4807 107'3-47C02V 106 110-26 1167li2.5 10 -2469,)03!4- 717694 43 1849 79607 6-667438613-603398 106 11-236 1191010 10 -29.56301 ,4 -73-2624 44 1936 861. HI 6 -633249ij!3- 630348 107 11149 1226043 10-3440804:4-747469 \> 2026 91125 6-708203;)l3-666y93 108 11664 1269712 10-392304«,4-702203 • UJ 21 Id 9; .-WO 0-7823300i3-6830Ks 109 llsSl 12950-29 10- 44030o6'4- 7768.06 ■' 4< 22i.i'J 10;5823 6-8666546!3-60c;8-20 110 12100 1331000 I0-48808o6i4-7914-20 •1.-! 2304 \W„'H 6 --.'282032 '3 -634241 111 12321 1367631 10-6366.533:4-805396 49 2101 117619 7-0000000 3-669300 112 12641 1404928 10 -.60300.02:4 -820-284 60 2600 12u000 7 -07 10678:3- 6840.) 1 113 12769 1442897 i0-630!463:4-834.6S8 61 200 1 132661 7-U11284!3-7081:;o il4 12996 1481614 10-6770783 4-848808 - 62 2704 I40o0w 7-2111026:3-73-2611 116 13226 1,520876 10-7233063 4-862944 b3 2;:0a Mo.ii? 7-2301099;3-76628i) 116 13466 1660896 10 - 77, ;3296 4-870999 -! ,j4 2;-il6 16,4o4 7-3484690 3-779763 117 1063'.) 1601613 10-816:) 538 4-890973 66 3U2;i 16637) 7-41619Su!3-80-2963 118 13924 hi 4303-2 10-8627306 4-904368 A .i^> 3i:>6 176616 7-48331 is;3- 826802 119 i-lOl 168^169 10-908712!:4-918686 .)7 3249 l.>361.93 7-6498341;3-843.601 1--.H) 1 ! 100 1728000 10-9644612 4-9324-24 k ;i:H 33(jt !9.M)-J 7-fi!6773!!3-.870877 i-2' 1 4641 177166! 11 -0000000 4- 94:>0a3 > 69 34t;l 2063;9 7 -681 14-57 Is -Ly 2900 122 1488 4 1816848 11-01.53610 4-969676 1 lit; 3G00 216000 7-746966713-91 1867 123 16129 1 860867 11-0110636)4-973190 1 (5! 3V2i 226'J81 7 -6102497 3- 93649 r 3-967802 124 16;i76 I;i0u624 11-1366287 4-986631 C2 3844 23.':!328 7-8740079 1-26 1602o 1963126 11-1303399,6-000000 ? 63 3989 260017 7-9372539 3-9790.57 126 1637f 2000376 ll-2-24972'i,6 -01 3-298 1 i'i ! l!f h J r; nil 'I j I ' M i 'I i ! ( 373 SQUARES, CI UKS, AND U00T8. No. 1Q7 Sqiinrt.i C'lilic. 130 lai 133 131 135 i:<6 137 138 139 MO 141 142 143 144 146 146 147 148 14!) 160 151 162 163 154 155 ISO 167 158 169 160 Itil 162 163 104 10.' 166 137 168 169 170 171 172 173 174 175 176 177 178 179 ttiO 181 182 183 1 84 18o! l8o| 187. 18H 189 1U120 I63H4 Itilitl 16900 17101 17424 176891 179661 18226 184961 187091 19014 193 Jl| 196901 198811 20164! 2(J44i» 20736 2102"; 21310 21009 21004 22201 22.>00 22?01 23104 2340!) 23716 240iJ.' 24336 24649 24964 36281 20600 26921 26244 26;j()9 26896 27226 276)6! 2788;) 28224 28061 28900 29241 29684 29929 302Vii 30026 3ii!i.'6 313J9 3168i 3J011 3-'4iJ0 32 ,"1)1 331 2048383 2097162 2146089 2197000 2248001 2290968 2362ti3 2400 1 U4 2lli03/., 2616 i:o 2671363 2028072 2686619 2MU)00Ul 2riO:U21'll 286.1'28,si 1 1 3:iis9 33860 34226 34.".96 3 1909 36341 36721 I 8q. lUiut. Ciitij lluul 11 -2694277 16-02H620 1 1-31 37086! 5- 0396S1 ir367H167i.»0627;4 ll-40i;643|6-066797 11 -4456231 1 6- 0787 63 ir4!i;91263|6-091613 ll-o3266-.'6|6 11-6768369 6 104169 117230 ■ 1-^9928 1 12603 •16613; 29:^ 1207 298,)98 4 30 (8626 3112136 3176623 32ll79i 3307949 337..000 344296! 3611808 3681677 366-2264 3723876 3796116 386i)»93 39443 1-i 4019679 40ll(i000 4173281 42616-28 4330747 4410944 4492126 6 189600 1 6 601903h;6 7046999:5 7473414 6-167649 7898261|6- 18010 8321696;.)- 192494 8713421 6 --204828 9l637.")3!6--217103 1 1 -968-260 7 1 6- 12-0000000 6 I 2-04169 lo! 6 12-08.'J0460'6 ^39321 211 183 263688 206 N.» 100 191 192 193 194 196 196 197 198 199 200 201 202 -203 ■201 20) 06 207 208 flquurc 637|-209 i2-1243.667|6'2776:i2l2' " 16.V)-i6l|6-289.<72 2066666 6 -.3014.-)9 12-2474487 6-313-293 1 2 -288206t;|6- 326074 12-3-2882S0 16-336803 12-3693169;6-318181 l2-409673i)!6-36010d 12 -4498996; 6 -37 1 686 12-4899960 6-383213 12-6299641 6-3i)4691 1 2 -.6638061 6-406120 12-6096-202, u-417601 12-6401106 6-4-28836 l-.'-6886776|6-44012 12-7279221 i6-461362 12-76714.63|6-462666 12- 806-2486 ;6 Cnbf. 12-84.")232rti5 46742..»6;12-8840987a 4667433! 12- 9228430 15 4M1032;12-961481 t 6 4826809, 13 -0900000 '6 4913000!i3-0384048"6 47.370-1 48481)0 496866 606879 617818 5287 7. '5 639668 6000211! 13 -076696»,6- 6^0 199 ■561-29,-; (7 206^1 ■ 604079 5088448 13-1148770 51777171 13-1629464 526 J024| 1 3 - 1 90.9060 6 - ,)82770 5369376 1 :', • -228766615 -f- 934 46 6461776: 13--266 4992:6 5.6462:j3J3-3041:)47i6 66:i;l7.)2; i 3 ■ 34 1 664 1 6 - 62.)226 6736339 1 3 - 3790882 5 - 6367 I i [>:;32O00 13-416 4(179 6-646216 6929711 13-46:i62.10;6-6-)6661 (;;)-28608' 1 3 - 4907376 6 - 667061 6 l-2'U87j 13 -6-277493:6 -6771 11 6229.JO 4' 1 3 - 6646600:6 - 687734 63;'.1626l.'.-601 '17066 -OOoOlO 6434866 13-638181715-708267 6.j39203 13-0747943;6-7 18479 66446 72: 13-711 3092;6 -728664 0751269 13-7477-271|o-738794 10 !11 212 213 214 215 216 217 213 219 220 221 222 223 224 -225 226 2-27 228 ■229 230 ■231 i 232l 233 234 236 236 237 2-W 239 -614673 210; 24 1 ; 242! 243 j 2441 246 1 246 247! •248 1 -249| 260 -251 1 252! 36100 36IB1 36864 37249 37636 38026 38416 38809 39204 39601 40000 40401 4t)804 41209 41616 42026 42436 42349 43264 43681 44100 44621 44944 45369 4.6796 46226 46666 47089 47624 47961 48100 48841 49284 497-^9 60176 50626 61076 51529 51984 6-2441 .52900 63361 53324 61289 54766 56-226 06696 56169 66641 57121 57600 58081 68664 69049 59636 60026 60616 01009 61.60-4 62001 tf2600 63001 63504 Sq. Iloot. Cub* Root 6859000113-7840488,5 09678711 13 •820-2750 5 7077888113-8561066 6 71890,17 13-8921440.6 7301384 13-n-283883 5 7414876 13-9012400 5 7620636 U'OOOOOOOl.5 76 4.6373 14-03.jfl888;5 776-2392 I4-07I2173'5 7880)99 8000000 81-20601 8242 108 8366427 848966 4 8616126 8741816 8889743 8998912 91233-29 9261000 9393931 9.6281-2S 9663697 9890341 993S375 10077696 10218313 10360232 10603469 10648000 10793861 10941048 11089667 11-2394-24 11390626 11.543176 11097083 11862362 12008939 12167000 12326391 12487168 12619337 1281290 4 1-2977875 13I442.J0 1331-2063 13431 27 J 13661919 13824000 13997621 1417-2-188 14348907 145-26789 14706126 14886936 16069223 1626-2992 16438249 15626000 16813251 10003008 11- 14- 14- 14 14 14 14 14 14 14 14 14 14 14 14 14 14- 14 U 14 14 14 14- 14- 14- 14- IS- IS- 16- 15- 16- 15- 16- 16- 16- I5' 15' 15' 16' IS- IS 16' 15 15 15 15 15 15 15 lo 16 15 15 15 1 067360 15 14213666 1774469:5 21-26704 5 2478068:5 2828669 i5 317821 1 15 35-2700115 387494615 42-22051 5 4,5683-2315 4913767 6258390 6602198 .59451it5 6237388 6628783 69693t 730<)199 7648231 7980436 8323970 8660637 899(i644 9331815 ■9666-296 0000009 0332961 -06fi6192J6 • 0996689 i.i 1327460 1657609 1986842 2315402 2643375 J6 • •297068616- .3297097 6 • 362-2916 6 - 3:11801316- 4-272486 16 - 4696248 '6 - 491933 I '6 ■ 624 1747 !6 - .6663492 i-ii - • 588 1673 16 • ■6-2019910- • 66-247 .68l6 • ■681387l|«- -7162336]6- -74b0ii>7;6- •7797338|6- •311388316- •84'29795 6^ •8745079 6- 748897 758965 768998 778996 788960 7P,S890 808786 818618 8-28476 838272 818036 -857766 867464 877130 886766 89636S 906941 916 ISl 924993 934473 943921 963341 96-2731 972091 931426 0907-27 OOOOOO 009244 018463 0-27660 ■036811 04.)913 0.66048 0641-26 •073178 -082201 -OOni;) -10017(1 -109116 •118033 - 126926 - 136792 -1 41634 •163419 -16-2239 •171006 -179747 13.J463 197164 20682 1 21446! 22 31 18 i 2;;i6?ii 2 40261 243800 267321 266826 274306 ■28076() 291194 299604 307993 316359 ,Cubi Rnot ,88 5-748807 .'>0,5'7.V<i»(l5 i6iJ6-7t5rf!W8 ,40.6-77H9!»0 isn 8 •7889(50 ,00&'7PSH90 I00l5-80H786 18H;5'8|8(U8 i73!5-8iSl7ti ltiOj5-83Hi7'i lu(iij'Hl8()3i) 100!o-8fl77ii(; 04 3-b67l<)1 Ib8|6-87713(» i69i»- 88076 "i !ll|5-896.'ttis )01io-90i)941 1-23 1 'ti' . 190 i 198 I li»5i 188 I f83i 186 I 199 I !;u 186 1 471) I )87 144 Slnl i'JJ I JO!) 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Rool. •27-6136.330 :27- 6317998 27-5499546 27-6680975 27-5862284 Cube Uotii 442450728 27-6043475 444194947 - 445943744 580225 447697125 586756 449455096 586-289 451217663 589824 591301 592900 594441 595984 597.529 774 599076 775 600625 776 002176 603729 605W4 606841 008400 609961 611524 013089 452984832 454756609 456533000 45.- ■^14011 460(»i, >648 401839917 463684824 465484375 467288576 469097433 470910952 472729139 474552000 476379541 478211768 48004808 27-6224546 27-6405499 ■27-6586334 27-6767050 27-6947648 27-7128129 117793 121801 125806 9-1-29806 9-133803 9-137797 9-141788 9-146774 9-149757 9-163737 9-167714 No.iSqn.-ire.l Cube 9-113781 820 |67240e|65I 368000 "•' ■■'^"- 821 1674041 155.3387661 Sq, Rool. {Cube Root 4855876") 28 -035691 5 4874434i»3 614656 48 1S90.3O4 616-225 4837366-*i 017796 619309 020944 622521 624100 625681 627264 628849 630436 032025 633616 27-7308492 9-161606 •i7- 7488739 9-165656 27-7668868 27-7848880 27-80-28776 27-8208555 27-8388218 27-8.567766 27-8747197 27-89-26514 ., 27-9105716 9 9-1696-22 9-173585 9-177644 9-181500 9-185453 189402 193347 197289 :01229 27-.Q284301 9-205164 27-9463772 27-964-2629 27-9821372 28-0000000 28-0178516 797 635209 " 3 036804 ) 038401 ) 040000 1 6U601 ! 043204 I 644809 046416 I 648025 ' 649636 051249 662864 054481 656100 657721 659344 813 600909 '^14 6«2596 31 -1664225 816,6058.56 817 489303872 491109069 493039000 494913671 49679'>'^88 498077 -io7 500566184 23-0535^203 •23-071.3.377 28-0,891438 28 -J 069380 28- 1247^222 •28-14-24940 23-1602557 23-1780056 502459375 28-1957444 504358330 28-2134720 506-26157.3 28-2311884 508169592 28-2-188938 51008-2399 23-2665881 512(JD0000 28-284-2,712 513922401 28-3019.434 515849008 28-3196045 517781027 28-3372546 818 819 019718464 28-3548938 021660125 28-3725219 023006616-28-3901391 520557943 28-4077454 527514112 28-4-253408 529475129 28-4429253 531441000 '28-4004989 53341173128-4780617 5353873-28 28-4956137 537367797 28-5131549 539353144 28-6306852 541343376 28-6482048 o4333«4»6 23-6657137 i,-<ytvioi 667489 54533851 3128 • 58321 1 919 - 343473 ----_-. ...,.,,.. ._..,..,^u3:tj;? ja;;»BD 670.oi|549353369J28-6181760 9-8»«O95 9-209096 9-213026 9-215950 9-2^20873 224791 228707 23^2619 23752S 240433 244330 2482.34 252130 256022 259911 !)• 263797 9-267680 9-271559 9-275435 9-279303 9-28317S 9-287044 9-290.907 •294767 •298624 302477 3063-28 810175 314019 9-317860 9-321697 9-325532 9-329363 9-333192 9-337017 9-340838 9-344667 822 675684 823677329 824678976 825J680625 820i682276 8271683929 828 1 685634 829 i 687241 8.30 688900 831 1690061 832|6922i24 833!693889 334690056 830 697220 830693390 8371700569 28 5504122-18 657441767 659476224 561515626 563559976 665609283 067663553 569722789 .571787000 673806191 576930308 578009637 680093704 582182876 584277056 - 586376203 838 702244 588480472 839 703921 590589719 840 706600 592704000 841 707281 594823321 842 708964 590947688 813 710649,599077107 844 712330 601211584 29 845 714025J603351126 29 846|715716j605495736 29 8'*7 717409 607645423 29 848 719104009800192 29' 3 19|720801 01 1960049 -29' 8o0 72-2500 014125000 -29- "" 29- -24201 725904 28 2.3 23 28 •m^ •28' •28' 28- 28- 28- 28 28 •28 23' 28' '28' 28- 28 28 28 '29 '29' -29- 9- 9- 9- 9- 9- 9- 9- 9- 9- 9-'; 9-^ 9^ 9- 9^ 016-295061 1.^., 6184702081-29 727609 620000477 729316 022835864 731025 6-260-26376 73-2736|027222016 734449|629422793 736164 631628712 737831 033839779 739600,036056000 741321038277381 743044 640003928 744769 746496 865748-225 042735647 044972644 647214626 866 749900 049461896 -"^^^.^ 807 751689 651714363 868 753424 663972032 869 755161 656234909 370|756e0O 65350.3000 758641 66077631 1 760384 663054848 762129 665338617 63376 667627624 65626 669921876 707370|672221376 709129 674520133 770884 6768.36162 879)772641679141439^5, 880)774400j631472000J29 Cioii770ioli0837y?»fjj J29 88-2^777924|686128968 29 871 872 873 874 875 876 877 878 9 29 29 29 29 29 29 '29' ■29- 29- 29 29 29 29 •29 •29 • 29' 29- •29- '29- 29 29 29 29 29 29' 29 ■6356121 !9 6530970 9 6705424 •6879766 •7064002 -7228132 -7402157 •7670077 •7749891 J9 ■7923601 9 ■8097206 ■8270706 -8444102 •8617394 -8790682 •8963606 ■9236640 9309023 9482297 -9654907 -9827635 -0000000 ■0172363 ■0344623 _ ■0516781 9 •0688837 9- 0860791 9 103-2644 9 12043969 •1376046 9 •1647595 9 •1719043 •1890.390 •2061637 ■2232784 ■2.103830 ■2574777 274.5623 2916370 •3037018 •3257666 •34'28015 •3698.365 ■3768616 ■3933769 ■4108323 4278779 4148637 4018.397 4788059 •4957624 ■5127091 6296461 6466734 -5634910 •5803989 •6972978 •6141858 ■6310648 „ ■647932519 66479391a 681644219 69848489 359902 303705 367005 371302 37.5096 378837 382676 380460 390212 394020 397790 401509 405339 409106 412809 410030 420387 424112 427894 431642 435333 439131 442870 440007 450,341 461072 457800 401520 465247 463906 472632 476396 480106 483813 4876181 491220 494919 498616 602308 •605998 ■609686 ■513370 517051 520730 524406 528079 531749 •535417 ■539082 5427.44 546403 550059 663712 •507363 ■561011 •664656 608298 671938 675574 679208. 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Hoot. 29 29 29 29 29 29 29 29 29 29 29' 29 29' 29- 29' 29- 29- 30- 30- SO- SO- 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30' 30' 30' 30' SO- SO- 30- 30- 30 ■7163159 ■7321375 •7489496 ■7657521 •7825452 9 ■79932S! 8161030 9 Cuba Root 8328678 •8496231 ■8663690 ■8831066 •8998328 '9105506 9332591 9499583 9666481 983.3287 •0000000 ■W 66620 •0333148 •0499584 •0666928 •0832179 •0998339 •1164407 ■1330383 •1490269 •166206319 •182776519 ■1993377 9 ■2158899 9 ■2.324329|9 ■248966919 ■26.5491919 2820079] 9 ■29851489 31501239 •33150189 ' 347981 :j 9 364452&,9 380915119 3973683 9 92 1|853776 788889024 935 8556-25]791453126 30^4138127 9 9-J6 857476;79402-2776 30^430'3481 9 937 8593-39 798597933 30-4466747 9 933|801184 799178752 30-4630924 9 939:863041 801765089 .30-4796013 9 930|804900 804357000 30^4959014 9 93l|86676180695449l!30^61229-36'9 932i868634;S09557568;30-5286750|9 933jS704b9.3121663.'i7j30-5450487 9 931|872350|814780504'30-C614136 9 935 874-2-35;8174O0375i30-6777697 9 936:876096, 830036856|30 • 6941 1719 937|877S69,S-33656953 30^6104657 9 938;879844| 635293672130 • 6-307857 9 939.t;8172l|837936019j30-6431069'9 '140 883600:830584000 30-6594194'9 41 885481.83.3237621 30-6757233'9 -593716 •597337 •600956 •604570 •608182 •611791 •015398 ■619003 •623603 •626201 •629797 -633390 ■636981 ■640569 ■644164 647737 ■651317 '654894 6.58468 663040 665609 669176 672740 676302 679860 683416 086970 ■090.521 •69406!' •697615 •701158 •704699 ■708237 •711772 •715305 ■718S35 ■722363 •725888 •729411 ■732931 ■736448 ■739963 •743476 •746986 •750493 •753998 •767500 '761000 '764497 •767993 771434 '774974 778462 '782946 785429 788909 79238e 796861 799334 Wo. S<|uaie 942 943 944 946 946 947 948 949 950 961 962 963 954 985 956 967 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 079 930 981 982 983 984 986 986 987 988 989 990 991 992 993 994 996 996 997 998 999 1000 887364 889249 891130 893025 894916 896809 898704 900601 902600 904401 906304 908209 910116 912025 913936 915849 917764 919681 921600 923521 925444 927369 929296 931225 933156 935089 937024 933961 940900 943841 944784 946729 948676 950628 952576 954529 966484 958441 960400 962301 964324 966289 968256 970225 972196 974169 976144 970121 980100 982081 984064 986049 988030 990025 99-3016 994009 996604 998001 1000000 Cubs. Sij. Root. Ciihe Koot 636896888 838501807 841232384 84.3908625 846590536 849278123 861971392 854670349 857375000 860085361 862801408 86.5523177 868250664 870983875 87372-2816 876467493 879217912 881974079 884736000 887603681 890277128 693056347 895841344 89363212.= 901428696 904231063 907039232 909863i*i<9 912673(».0 915493(),1 918330C48 921167317 924010424 926869376 9-39714176 932574833 935441362 938313739 94119-2000 944076141 940966168 949862087 952763904 956671626 958636256 961504803 964430272 967361669 970299000 973-342271 970191488 979146657 982107784 985074875 938047936131 991026973131 994011992131 997002999 31 100000000031 30 30 30 30 30 30 SO SO 30 SO- SO- SO- SO- 30- 30- •St 30 30 30 31 31 31 31 31 31 31 31' 31' 31' 31' 31 ■ 31- 31- 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31 31' 31' •6920185 •7083051 '7245830 ■7408523 7571130 •7733651 •7896086 •805ai36 ■ 8-320700 •8382879 •8544972 •8706981 ■8368904 •9030743 ■'■ 193497 9354166 •9515761 ■9677251 •9838663 0000000 0161248 0322413 0483494 0644491 0805405 0966236 1126984 •1287648 •1448230 ■1608729 •1769145 •19^39479 •2089731 •2249900 •2409987 •2669992 •27-39916 •2889787 •3049517 •3209195 •3368792 •3528308 ■3687743 •3847097 •4006.309 •4165561 •4324673 •4483704 ■4643664 ■4801525 ■4960315 •6119025 •6377665 ■5436206 5594677 '.>r.5.3068 6911380 6069613 6227766 9 9 9 9 9 9 9 9 9 9 9 9 9' 9- 9- 9^ 9 9 9 9 9 9 9 9 9 9 9 9 9- 9- 9- 9- 9- 9- 9^ 9^ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9' 9' 9- 9- 9- 9- 10- •8O3804 '806'r/l 80973O 813199 •816655 •820117 •8-33572 •8-37035 •830476 •833934 •837S69 •840313 ■844234 •847692 ■851128 ■854562 857993 •8614-32 •864848 •868373 •871694 •876113 •878530 •881945 •885367 •888767 •892175 ■895580 ■898983 ■902333 •906782 •909178 912671 916962 919351 922738 926122 •9-39504 •932884 •936261 •939636 •943009 •946380 •949748 •9.531141 •956477 •959839 •963198 -9665,55 •969909 ■973363 ■976U12 •979960 983305 9866 H) 989990 993339 9%'666 000000 Nu. (iij \iy. 1 2 3 4 5 6 7 8 9 10 11 13 13 14 1,5 l(i 17 19 30 21 33 T.i •Ji I Sij. Root. Ciihe Koot M- 6920 185 30-7083051 30-7245830 30-7408523 30-7571130 30-7733651 30-7896086 30-8058-136 30-8320700 30-8382879 30-8544972 30-8706981 30-8868904 30-9030743 30 •'•192497 3* 9354166 30-9515751 30-9677251 30-9838668 31-0000000 31-0161248 31-0322413 31-0483494 31-0644491 31-0805405 31-0966236 31-1126984 31-1287648 31-1448230 31-1608729 31-1769145 31 -19-^9479 31-2089731 31-2249900 31-2409987 31-2569992 31-27-29915 U -2889767 11 -3049517 (1-3209195 11-3368792 11-3528308 11-3687743 11-3847097 (1-4006309 (1-4165561 H -4324673 il -4483704 H -4642664 il -4801525 1 -4960315 1-6119025 1-6277665 1-5436206 1-5694677 1 -.1753068 1-6911380 1-6069613 1-6227766 9 9 9 9 9 9 9 9 9 9 9 9 9- 9- 9- 9- 9- 9- 9 9 9 9 9 9 9 9 9 9 9 9 9' 9 9' 9- 9- 9- 9- 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9' 9' 9- 9- 9- 9- 10- •8O'.'M04 - 8062V 1 -809736 813199 -8I66.-)5 -82011; •823072 •827025 •830476 ■833924 •837369 •840813 •844254 847692 851128 854562 -857993 -86 1422 -864S48 •868272 -871094 -875U;} •878630 •881946 •885367 •888767 •892175 -895680 •898983 ■902383 ■906782 ■909178 912671 916962 919351 922738 926122 •929504 •932884 •936261 •939636 •943009 •946380 •949748 •963114 •966477 •959839 -963198 -966555 ■969909 •973262 ■976012 ■979900 983305 986649 989990 993329 996666 000000 i TABLES. 3S'> •^'^^ ^^- ^^ T"^^- AMOU NTS OF £1 AT COMPOUND INTEREST. 3 per cent Mo. of I'ay- mcnt] 4 per cent 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 26 per cent 1-03000 1-00090 1-09273 !• 1-255 1 1-169'27 1-19405 1 -22987 1-26677 1 -30477 P343!42 1 •38423 1 •4-2576 1 •4U853 1 •51259 r 65797 1^60471 1 •65285 70243 75351 806 11 860-29 91 610 1-97359 2-03279 2-09378 l: 1- 1^ 1- 1- 1- !• 1- 1- 1- 1- 1- 1- 1- 1- 2- 2- 2- o. 04000 08100 1-2486 16986 21063 26532 81593 36S5T 4-2.331 48024 53945 ■60103 06507 73168 80094 87298 94790 02582 10685 19112 27877 2-3o!Jri2 2-40472 2-66330 2-60584 1 •05000 !• 10250 1^ 16762 1 •21551 1^276^28 !• 34010 1^40710 1-47745 1^56133 1 •62889; I • 71 034 1-79586 1-88565 1-97993 2-07893 2-18287 2-29202 2-40662 2 •5-2695 2-65330 2-78596 2-925-20 2-07152 3-2-2510 3-38635 6 per cent •06000 •12360 •19102 -26248 ■338-23 •41852 ■50363 59385 68948 Vo. of Pay meiita 1-79086 -89830 •012-20 •13-293 •26090 •39656 ■54035 •69277 85431 02560 20713 39956 3-60354 3-81975 4-01893 4-29187 20 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 43 49 60 3 per cent 2-16659 2-22129 2-28793 2-35657 2-42726 2-50008 2 '57508 2-05233 2-73190 2-81386 2-89828 2-98523 3-07478 3-16703 3-26204 3-35990 3-46070 •56452 67145 78160 89504 01 190 13-225 25622 ■i per cent 4-38391 77-247 88337 99870 11865 24340 37313 3 •50806 3-64833 3-79432 3-94609 10393 26809 43881 616371 80102 99306 19278 40049 5-61651 6-84118 6-07482 6-31782 6-57053 6-83335 7-10663 5 per c«nt 6 5 6 6 6 7 !■ 7- 8- 8- 8- 9 9 10 10 11 -56567 -73346 -9'2(ll" -11614 -32191 -53804 ■76494 •00319 25335 51601 79182 •08141 •38548 •70476 •03999 •39199 •76159 •14967 •5671.' •98,501 434-26 90597 40127 92183 46740 6 per cpiil 6 6 7 7 8 8 9 9 10- 10- 11^ 12- 12- 13 14 16 16 17 18 ■64933 8-2235 ] 1 11)9 41839 74319 •08.^10 •453;t9 •810:>LI ■25 1(1 2 '6StUtJ I 1725 fi3(l(rj 15125 70351 •28572 ■90280 •55703 •'23045 ■9.3543 ■76401 ■59049 •46592 39387 37750 42015 J'ABLE OK THE AMOUNTS OF AN ANNUITY OF £1. Su. (ilj 1 P.iy. Spercenl. <pcrci;iit| operctfiit 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 l(i 17 IS 19 20 21 22 •23 •2i I 6 7 8 10 11 12 14- 15' 17- 18- 20- 21- •23- 25 :i) 28 )0 !2' il- -oonoo •03000 ■09l)i)0 -J 8363 -30913 •40841 • 66246 •89231 ■15911 ■463t*S 80779 19203 61779 08632 59H!)! !M-!8!^1 7(il.i'.tl-23 'iU13l25 lH>'i7J27 •"'7(!37 -29 OTorsUi 53{i7Sj34 4.V2s,-ii3(i 4-6l7J3:l 4^y20'41 1 2 3 4 6 6 7' 9' 10- li- 13- 15- 16- 13 20 -00000 ■((4000 -12I60i •■24610 -41032 -63297 ■89829 •21423 58279 00611 4vS63; 1 6 per eentl' 00000 05000 15250 3 31012 4 62563 5 80191 6 14201 8 9-54911 9 11-0-2656 11- 12-57739 13- M- 20679 14- 02580 15-91713 16- 6-26Sl|l7^7i29!i 18- 29191 |19*?)9S63 21 • 02359-21 •o7S.-i6 23 • 3 per cent ■« per cent 6 per cent •S-2J.03''23-65749 ■69751 25-84037 ■61541.-28-I323S 671-23 00-53900 778(»,M 33 •06595 96920 35 •710-.).:, ■2171)7 ,{8-50521 0I7.'-D41-.130!/ (lo'iiid ■I4^5(12U0 25 -( 28--. 30-f 33-7 36-7 39 - P 43 -S fti-;i .50 ■ S 6-loWl 4V-,;.'71tli54 50 18-66304 SO -70963 2-93092 5-21885 7-57641 0-00268 2-50276 5-07784 7^73018 »• 46208 3-27594 3-17422 M5945 2-23423 >-40126 ?• 66330 !-02320[l04 '•48389 110 I-048U 115 !-719y6 121 ■•50146'l26^ • 396.50 '132 - • !0.S39,139- •5106,V|I5- ■796tt7jl5'2 44 47 49 52 66 69 62 66- 69- 73 ■ 77 • 81^ 85- 90 95 99 •31174 •08421 •96758 •966'29 •08494 •32833 ■7QU7 ■20953 85791 65-222 59831 70225 61 64 68 62 66 70 75 80 85' 90' 95- 101 •11345 •60913 •4025S •32271 •43886 •76079 •29329 •06377 ■0669t; 32031 8363i' 628 M 6 percent o9 63 68 73 79 81 90 97' 104 111 119 J 27 97034| 107 •70954 135 •09502 145 • •79977 154 • •8397(;i65^ '•23175 175- ■99334 187 114 120 127 •4091 ■02551 ■82654 ■31960|l35 01238 142 412881151 02939J159 87057 168 94539 178 ■2632i|l88' 53.';73:li)S 667081 209- I 14300 •7001(.. ■68511 ■11942 025.3'> 42(iGu a47Pii 199 212 22(i 211 256 ■?2 00 •15638 •70576 •62311 ■03980 ■05819 ■S0168 88978 •34316 •18375 •43478 •12087 •26812 ■904-20 •05846 •76196 04768 95054 •6075e ■76303 •74351 j ■50812 09861 56453 95840 33590 Ml III 386 TABLES. TADLR OF THE PRKSKNT VAM/F'.S OV AN AN'XITITV -Vo. of IMy- I 3 a 4 6 6 7 8 9 10 11 li 13 14 IS 16 17 18 19 20 21 22 23 34 33 OF £1. 3l'erc«iit j<i,trcBm f' larceiitl i)»r cant 0-970&7 l-!)i;)-ir 2-8:28(il 3-7J71() 4-Wt)Tl a-41719 7 -01 oaf) 7- 78611 8- 53030 9-95400 10-63490 U -29607 U- 93794 12-66110 13- 1661-2 13-76351 14-323S0 14-8774,-! 15 •41602 15-93692 16-44361 16-93664 17-41316 o-aois-i l-8t-i6l9 2-77519 3-6^999 4-4dl)^i 5-a4214 6-00-.>0o fi- 73274 7-43533 8-1J089 8-76058 9-38507 9-98565 10- 5631-2 ll-11849fl0 11-65239 10 12-10567 12-65940 13-13394 13-59032 14-02916 14-45111 14-85684 15-24696 15 •6-2208 -95238 -85941 -75326 -54595 -32948 -07569 •78G37 •46321 • 10782 •72173 30641 86325 3SI367 89864 0-94310 1-83339 2-67301 3-46510 4 -21 236 4-91732 6-58238 6-30979 6-80169 7-36009 7-886a7 8-38394 8-&-V268 9-29498 N'o. nt l>ny- ■i per cent •< per car.'. 37965 9-71-225 83777110-10589 27406 -68968 •08533 •46-221 •82115 10300 48357 79864 09394 10-47726 10-82760 11-15811 11-46993 11-70407 12-04158 13-30338 12-65036 12-78335 26 37 28 39 30 31 32 33 31 35 36 37 38 39 40 41 43 43 44 45 46 47 48 49 60 17 IS 18 19 19 30 •20 30' 21- 31- 31- 33- 33- 33 33 33 33 33 34' 34' '24- 35- 35- 35- 25 •87684 32703 76411 -18346 •60044 -00043 -38S77 •76579 -13184 •48722 ■83-225 •16724 49246 80822 11477 41-240 70136 5 per cent I 6percfu» lt>- 98277 16-32968 16-^6306 16-98371 17-29303 17-08849 17-87356 18-14704 18-41119 13 •66461 IS •908-28 19 •I 4258 19-36786 19-68448 19-79277 19-99305 20-18563 93190-20-37079 30-64884 30-7-2004 20-88465 21-04393 21-19513 31-34147 31-48318 -354-28 -51871 •77545 ■ 02471 •36671 60166 73977 14-37518 14-64303 14-89813 15-14107 16-37-245 15-59381 15-80367 16-00255 16-19290 14 163741S 14 16-64685 16-71138 16-86739 17-01704 17-15908 17-29436 17-42320 17-64591 17-66377 17-77407 17-88006 17-93101 18-07715 18-16873 18-35593 14 14 14 14 15 15 16 15 16 15 16 15 16 16 15' I •00316 ■31083 40616 69073 76483 93908 08404 33033 •36814 •49824 -62099 •73678 •84603 ■94907 ■04630 13801 32454 30617 33318 46583 62437 68903 66002 70767 76186 IRISH CONVERTED INTO STATUTE ACRES. Iriah. SRltUU. R. P. u 10 30 1 2 3 A. n 1 3 p. 1 3 4 6 8 16 32 r. n 26 Ml 3 6 13 Iriih. 34 24 9 172 1 34 ll| A. 1 2 3 4 6 7 S 9 10 Statute. Iriih. A. K. p. T. A. 1 3 19 6 20 3 38 10 30 4 3 17 153 40 6 1 36 21 50 8 15 26.1 ij 6j 111 100 9 3 35 200 11 1 14 300 12 3 33 400 14 2 12 17 600 16 31 2-2i 1000 Statute. A. B. F, T 32 1 23 14} 48 2 16 6i 64 3 6 28/ 80 3 38 20l 161 3 37 101 323 3 34 2l| 485 3 32 2 647 3 29 13| 809 3 26 23 1619 3 13 163 VALUE OF FOREIGN MONEY IN BRITISH, Silver being 5*. per ounce 1 Florin is worth 16 Schilliiig.s (Hamburg) 1 Mark (Frankfort) . 1 Franc 1 Milree (Lisbon) 8 Reali . . ». d. 1 8 15| 9i 4 8 3 IJ 1 Dollar (Now York) . 96 Skillings (Copenhagen) 1 Lira (Venice) 1 Lira (Genoa) 1 Lira (Leghorn) . 1 Ruble . . *. d. 4 3 2 2| 8} 3 TV OF £1. 5 per cent 6perciu(l 14-37618 14-64303 14-89812 15-14107 16 •37-245 15- 69-281 15-80267 18- 00255 16- 19290 16-3741S 16-61685 16-71128 16-86789 17-01704 17-15908 17-29436 17-42320 17-64591 17-66277 17-77407 17-88006 17-98101 18-07715 18-16872 18-25592 13-00316 13-21083 13-40616 13-69072 13-76183 13-9-2908 14-08404 14-23023 '14-36814 14-498-24 14-6209«» 14-73678 14-84602 14-94907 16-046,10 16-13801 16-22454 15-30617 16-38318 16-45583 16-62437 15-68903 16-66002 15-70767 16-76186 Statute. A. B 32 1 48 2 64 3 80 3 161 3 323 3 485 3 647 3 809 3 619 3 P. T 23 14} 16 6.1 6 28j 38 20| 37 10| 34 21| 32 2 29 123 26 23 13 16J ». d. , . 4 3 1 jcn) o 2 . . U 8 . , 9 • . 7 « . 3 li